issues in multirate process control - University of Alberta

ISSUES IN MULTIRATE PROCESS CONTROL Arun K. Tangirala1, Dongguang Li1, Rohit Patwardhan1 Sirish L. Shah1 and Tongwen Chen2

Department of Chemical and Materials Engineering, University of Alberta. Edmonton, Canada, AB T6G 2G6 2 Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada, AB T6G 2G7 1

Abstract

Multirate systems are encountered when some signals of interest are sampled at a dierent rate than others. For example, in the process industry, composition measurements in distillation columns are typically sampled at a slower rate than temperatures and ow rates. In the context of closed-loop control, such multirate systems pose a challenging problem due to several reasons such as increased complexity in the design with tighter performance speci cations. Lifting techniques provide a suitable framework for posing a multirate univariate/multivariate problem as a multivariable single-rate problem. In this work, we discuss the application of lifting techniques with respect to asymptotic setpoint tracking. Theoretical results are provided to show that there are constraints on the controller gains for step-type reference signals to ensure there are no intersample oscillations in the closed-loop system. Discrete lifting usually introduces non-uniform steady-state gains for the open-loop lifted model which could result in oscillatory continuous output signals for the closed-loop system. These results are supported by simulation results of a slow sampled and fast control system. Further, we provide a continuous-time approach to design of multirate controllers while providing benchmark for comparing the closed-loop performance of multi-rate and single rate systems in the LQR framework.

Keywords:- multirate, lifting, control

1

Author to whom all correspondence should be addressed - email: [email protected]

1

1 Introduction Multirate systems occur when signals comprising a system are sampled at dierent rates. This situation is likely to occur when it is not possible to sample all the physical variables of a system at a single rate. In chemical processes, for example, the ows and temperatures are typically sampled at a faster rate than variables such as compositions. One would expect that multirate controllers would give better performance than single-rate controllers. For instance, these controllers would be preferable to the single-rate controllers because of the extra degrees of freedom they allow in manipulating control variables. This and several other factors have motivated researchers over four decades (Kranc, 1957; Kalman and Bertram, 1959; Crochiere and Rabiner, 1983; Meyer and Burrus, 1990; Ravi et al., 1990; Chen and Francis, 1991a) to develop techniques to handle these complex yet appealing systems. Issues dealing with design of optimal multirate control are also discussed in (Araki and Yamamoto, 1986; Meyer and Burrus, 1990; Ravi et al., 1990). Often, in practice, we would want to emulate the continuous system for reasons of better performance. This motivates the need for fast discretization. Moreover, faster control moves might be necessary to achieve an improved performace. Intersample rippling is an important problem especially because it re ects the continuous-time performance of the system. However, limitations exist on the sampling rates of certain variables in many systems. In single rate system, the slowest sampled variable poses limitations on the rate at which the control updates can be made. However, for a multirate system, control updates can be done at a faster rate. Thus, for a multirate system, it is appealing to build a controller that can provide faster input moves based on the less frequently available outputs. Traditionally, extended Kalman lters (Kalman, 1960), have been used in the multirate framework (Glasson, 1983; Gudi and Shah, 1993) along with the use of inferential models of the process. When the outputs of concern are not available at regular sampling rates, other measurable secondary outputs can be used with an inferential model to obtain estimates of primary outputs (Gudi and Shah, 1993). These estimates with an appropriate control law would give the control updates. A multivariable, multirate self tuning control algorithm has been implemented on a distillation column by Tham et al., (1991) where they have shown that improved performance can be obtained in comparison with a PI=PID based strategy. In this work, however, a dierent approach is given where we do not consider an inferential model. Instead, control updates are made at the faster rate using lifting techniques, while the measured outputs are only available at the slow sample rates. The open-loop lifted model of the multirate system contains more information than the model available at the slow sample rate. The comparison of control performance of multirate versus single-rate using a suitable benchmark is also an open problem. The performance assessment of multirate and single-rate systems based on a continuous time LQR objective function wherein we consider a regulatory control problem is considered in this paper. The continuous-time cost function is shown to provide a fair benchmark for comparison of multirate vs. single-rate systems. Theoretical analysis and simulation show that there are some constraints that need to be satis ed by controllers based on the lifted model for asymptotic set-point tracking. In this work we analyze these limitations and a way to overcome them. In the second section, we review the application of lifting techniques to multirate control. The third section deals with the theoretical analysis of the limitations of multirate controllers based on lifted models and presents a solution to overcome them. In the fourth section, we extend the results given in (Chen and Francis, 1991b) to obtain an equivalent single-rate discrete system based on a continuous-time LQR cost function when applied to multirate systems. These results provide a continuous-time approach to design of multirate controllers. The paper ends with some concluding remarks and suggested future work. 2

2 Multirate Control Using Lifting Techniques In this section, we brie y review 'lifting' and discuss its application to multirate control from a chemical engineering perspective. For an elaborate discussion, the reader is referred to (Chen and Francis, 1995). Lifting techniques essentially are the result of concatenation of signals resulting in the transformation of multirate single-input single-output (SISO) or multi-input multi-output (MIMO) systems to a single-rate MIMO system. Such a transformation is very useful since all of the theoretical results for single-rate systems can be applied to multirate systems.

v

- Hf

- G

p p p p p p p p p p p p p p p p p p p

- S

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p p p p p p p p p p p p p p

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v - ,1 L p

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- Hf

p p p p p p p p p p p p p p

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- Sf - Hf p p p p p p p

- S

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(b) Figure 1: (a) A multirate system (b) Single-rate lifted system (high frequency dots indicate fast-rate signal , low frequency dots indicate slow-rate signal and solid line corresponds to continuous signal) Consider a discrete SISO system where we have inputs v coming in at a rate 2h and outputs  at a rate 3h. The system comprising these discrete signals v and  could be periodically timeinvariant with a period of 6h. Since these signals are available at dierent sampling instants, it is not possible to have a transfer function between v(k0 h) and (kh) (k0 = 6 k) where v and , the

input-output pair, are represented by

v = fv(0); v(2h); v(4h); v(6h);


g

and  = f(0); (3h); (6h);


g

However, let us consider these set of signals over a period of 6h (the Least Common Multiple (L.C.M.) of 2h and 3h ). A slight rearrangement would give the following representation

82 3 2 3 9 > = v = >4 v(2h) 5 ; 4 v(8h) 5 ;


> : v(4h) ; v(10h)

3

and outputs

=

("

# " # ) (0) ; (6h) ;


(3h) (9h)

A system comprising these set of inputs and outputs is evidently time-invariant and it is also clear now that the original system is periodic with a period of 6h. Hence a transfer function form can be obtained for this system with these multivariable transformed signals. This transformation of signals through which we have obtained a time-invariant system from a periodic, time-variant system is an example of what is called lifting. The new set of signals which we have obtained are referred to as lifted signals. This operation, as we have seen, resulted in the transformation of a multirate SISO system to a single-rate MIMO system by just rearranging the signals in a particular fashion. Extension of these ideas to multirate MIMO systems is straightforward (Chen and Francis, 1995; Chen and Qiu, 1994). With a time-invariant system in hand, the obvious next step would be to obtain a model based on these set of signals. It is possible to obtain such a model based on either the continuos-time domain model or a fast-rate discrete-time model. In this study, a multirate system is considered where the inputs to a linear time-invariant (LTI) plant G are available at a faster rate than the outputs. The setup for such a system is shown in Fig.1(a), where S and Hf refer to the slow sample with a period of hb and fast hold with a period of hb =n respectively, hb being the base sampling period (the slowest sampling period in the system). In the following analysis, it is assumed that the outputs are sampled at an integer multiple n of the faster rate. Other cases of multirate systems can and do exist, e.g. fast sampling and slow control. However, the former case is of more practical importance in the chemical process industry. Formally, the term `lifting' refers to the linear tranformation Ln : l(Rm ) ! l(Rmn ), having certain properties. Consider a discrete signal v 2 l(Rm ) obtained by sampling a continuous signal u(t) (where dim(u(t)) 2 Rm ) at a rate hb =n, hb being the base sampling period (slowest sampling period in the system) and n a positive integer. Lifting this signal v results in (dropping the subscript on L) 82 3 2 3 9 > > v (0) v ( n ) > = 77 ;


v = Lv = >66 .. 77 ; 66 .. . . 4 5 4 5 > > > : v(n , 1) ; v(2n , 1) where v = fv(0); v(1); v(2);


g Note that v has an underlying period of hb , the base sampling period. Some of the useful properties of the lifting operator L are listed below : 1. As a system, L is non-causal and time-varying 2. L is norm-preserving : kLvk2 = kvk2 , where k:k2 indicated the 2-norm for discrete signals. 3. The inverse lifting operation L,1 de ned below is causal, but time-varying, where

v = L,1 = f 1(0);


; n (0); 1 (1);


; n(1);


g 82 3 2 3 9 > > 1 (0) 1 (1) > = = >66 .. 77 ; 66 .. 77 ;


> 4 . 5 4 . 5 > > : ; (0) (1) n

n

4

4. LL,1 = I and L,1 L = I From the above discussion, it can be seen that lifting results in mapping a fast-rate signal to a slow-rate signal with an increased dimensionality. While this operation maps a fast-rate signal to a slow-rate signal, the inverse operation maps a lifted signal to a fast-rate signal. The matrix representations of L and L,1 are given in (Chen and Francis, 1995). From the discussion we had earlier, we modify Fig.1(a) to Fig.1(b) to get a time-invariant system. Inputs to the controller built on the lifted system represented in the dashed box in Fig.1(b) would consist of slow outputs in the system, while the controller outputs would be the lifted input signals to the system. The lifted signals have to be thus inverse lifted and passed through the fast hold Hf . The design yields an LTI controller whose system matrix is block lower-triangular and causal. Referring to this lifted controller, the causality constraint requires that the feedthrough terms must be block lower-triangular - see for example (Chen and Qiu, 1994). In (Meyer and Burrus, 1990) and (Ravi et al., 1990), the causality constraints are discussed at length. Recently, Chen and Qiu (1994) proposed a new framework for dealing with multirate systems while dealing with the causality constraint. By our earlier assumption of the input being sampled at an integer multiple of the output sampling rate, this causality constraint does not arise. State-space techniques, both for modeling and control, provide a good insight into the application of these methods and are also a convenient tool for handling multivariable systems. In order to apply lifting techniques, we rst need to obtain a state-space model for the timeinvariant system. For this purpose, the following equations may be followed from Figure 1 and from the results developed in (Chen and Francis, 1995)

SGHf = SGHf L,1L = SHf Sf GHf L,1 L = [|I 0 {z


0]} LGf L,1 L n blocks z }| { = [I 0


0] GLL = G~L L where Gf = (Af ; Bf ; C; D) refers to the step-invariant fast discretization of the continuous-time system G and GL = LGf L,1 is called the lifted system (Chen and Francis, 1995). G~L is a timeinvariant system and hence a transfer function exists. Using the continuous-time state-space model for a stable plant G " # A B g^(s) = C D we get the time-invariant system G~L , " n n,1 n,2 B


B # A B A A f f f f f f g^L () = C D 0


0

(1)

Thus, we can easily obtain a state-space model for this time-invariant system by simple lifting techniques. It is very important to note that though there is an increase in the input dimensions, the order of the state-space model is preserved. This property makes this technique more appealing as the optimal controller based on the lifted system will still have a maximum order not exceeding the model order. Once we have this single-rate state-space model in hand, the implementation is straightforward. 5

3 Limitations on the multirate controller Lifting of discrete signals increases the dimensionality of the problem, while enabling all the lifted signals to be available at the same rate. The second feature provides the actual basis for converting a multirate problem into a single-rate problem. In a control design problem, the original system is lifted suitably and then a controller (we call this a multirate controller) is designed to compute the lifted input signals based on the available lifted/original outputs. In this case, the outputs are not lifted and hence the controller gives the lifted input signals based on the slow-sampled output. The outputs of the controller are basically all the input moves between the sampling period of the outputs. However, we have observed theoretically and through simulation that there exist limitations on the design of such controllers which result in intersample oscillations in the input-output signals. In this section, we present these limitations for the speci c case of slow-sampling and fast-control multirate problem and the solution to overcome these limitations. These limitations should not be confused with the causality constraints on the controller which exist for other types of multirate systems.

3.1 Analysis of gains

Consider the above mentioned SISO system where we have inputs coming in at a rate with period hb =n and outputs going out at the rate hb . Denoting u(k0 ) = Shf u(t), x(k) = Shb x(t), y(k) = Shb y(t) and u(k) = [u(k0 ) u(k0 + 1)


u(k0 + n , 1)]T , we have the following state-space model for (y; u)

x(k) = Anfx(nk) + [Anf ,1 Bf Anf ,2 Bf


Bf ]u y(k) = Cx(k)

(2) (3)

or in a transfer function form as

y() = C (I , Anf),1 [Anf ,1 Bf Anf ,2 Bf


Bf ]u() These n transfer functions relating the output to the n lifted signals can be re-written as :

GL;1() = C (I , Anf),1 Anf ,1Bf GL;2() = C (I , Anf),1 Anf ,2Bf

.. . GL;n() = C (I , Anf),1 Bf

(4) (5) (6)

where we have assumed a strictly causal system G (D = 0) and considered the frequency domain  = z,1 for convenience. This result can be easily extended to MIMO systems as well. Evidently, G1(1) 6= G2(1) 6=


6= Gn(1) which implies non-uniform steady-state gains for the lifted system and thus can be the cause of intersample ripples. The above result implies that for set-point changes, the nal steady-states of the n control moves obtained from the controller are not identical. When these lifted inputs are inverse lifted and passed through a zero-order hold, the continuous input to the plant at the steady-state is not steady. Note that for a unit set-point change, the n control moves would be just the pseudo-inverses of the steady-state gains of the lifted system. For disturbance changes however, these oscillations are not observed. For set-point changes since the nal values of the lifted inputs are dependent on 6

d

Fast Hold

Process

sample rate j T s

JTS

Inverse Lifting

Lifted Controller

r

Figure 2: Multirate system: slow-sampled fast-control system (J=j = n) the steady-state gains of the lifted system this problem is observed. In a way, this problem sets in when the lifted inputs are inverse lifted. Consider, for example, a SISO system with n = 2. If we denote the controller output (lifted inputs) in time as (assuming the fast sampling period as 1 time unit) as : (" # " # " # ) u(0) ; u(2) ;


; u(2ks ) ;


u(1) u(3) u(2ks + 1) and further assume that after the ksth instant the signals reach their individual steady-states, then inverse lifting of the above signal would result in the following sequence

fu(0); u(1); u(2); u(3);


; u(2ks ); u(2ks + 1);


; g Thus after the ksth instant, due to dierent steady-states of each of the lifted inputs, oscillations would set in after inverse lifting. It follows that the continuous output is oscillatory as a result of the continuous oscillatory input. However, the discrete output may not exhibit any oscillations. Observe that these oscillations are n-periodic.

3.2 Limitations on the controller gains

We discuss here the constraints on the controller gains for asymptotic tracking of step type reference signals based on the theoretical analysis of the intersample outputs. For stable systems and for step-tracking, it is known that for ripple-free continuous output, we need a constant input signals (Franklin and Emami-Nacini, 1986). This implies, in this case, the fast-rate or the inverse lifted signals be constant at steady-state which in turn implies that at steady-state the lifted signal u = [ui ]in=1 has all ui 's identical. Denoting the controller as CL in Fig.7 there exist n transfer functions between the error signal and the controller outputs, Cl;1 , CL;2 ,


, CL;n . Hence,

7

Also, from equations (4) and (6)

2 CL;1()e() 66 CL; ()e() u() = 66 2 .. 4 . CL;n()e()

3 77 77 5

(7)

y = [GL;1 GL;2


GL;n]u

where y and u are the output and lifted inputs respectively. The following theorems are stated for stable SISO systems. Note that if the fast-rate plant Gf is stable, the lifted system GL is also stable. Theorem 1 Consider a stable closed-loop system comprising a discrete multirate controller CL and the lifted system GL = SGHf L,n 1 consisting of the lifted inputs u = Ln u and output v, where n is the slow-to-fast sampling ratio and u is the fast rate input signal. Then, if Gf is strictly stable, for step type reference signals, the following condition needs to be satis ed to ensure that no intersamples are present in v. CL;1(1) = CL;2(1) =


= CL;n(1) where CL;i (1) i = 1;


; n are the gains of the n controller transfer functions. Proof: For the closed-loop system, we can write

ui() = CL;i()e() = For set-point changes,

CL;i r() n X 1 + CL;i GL;i

(8)

i=1

r() = 1 ,1 

If GL is stable, we have nite gains for the lifted system and using the nal value theorem, it is straightforward to show that CL;1(1) = CL;2(1) =


= CL;n(1) is necessary for all the ui to be equal. If CL;i has an integrator, then a similar condition holds. In such a case, we can write C 0 () CL;i = 1L;i,  (9) 0 () to have a nite gain. where we assume CL;i Substituting equation (9) in the closed-loop equation (8) for ui and using the nal value theorem again, we arrive at the following result 0 (1) CL;0 1(1) = CL;0 2(1) =


= CL;n

2

Thus, we have shown for a stable SISO system, even in the presence of controllers with in nite gains need not necessarily eliminate intersample ripples in the output for step type reference signals. However, a fast rate integrator present in the plant can ensure steady-state output at the fast rate. 8

Theorem 2 Consider a stable closed-loop system comprising a discrete multirate controller CL

and the lifted system GL = SGHf L,n 1 consisting of the lifted inputs u = Lnu and output v, where n is the slow-to-fast sampling ratio and u the fast rate input signal. S is the sampler at the slow-rate hb and Hf , the zero-order hold at the fast-rate hb=n. Then, if Gf is strictly stable and augmented with the lifted integrator GLI = LnGIf L,n 1 , where GIf is the discrete integrator at the fast rate, for step type reference signals, there are no intersample ripples in the steady-state output . Proof: Augmenting GL with the lifted integrator GLI is equivalent to lifting the system obtained by augmenting the fast-rate plant Gf = Sf GHf with the fast-rate discrete integrator GIf . First, we note that GL = SGHf L,n 1 = SHf Sf GHf L,n 1 = SHf Gf L,n 1 GLGLI = SHf Gf L,n 1LnGIf L,n 1 = SHf Gf GIf L,n 1 = SHf GuL,n 1 where Gu is the augmented state-space model of Gf with GIf and we have made use of property 4 of lifting. Let us denote the poles of the fast-rate discretized plant Gf as p1 ; p2 ;


; pm (need not be distinct). With the augmentation of an integrator, there is an additional pole introduced to Gf at  = 1. Thus the augmented plant has poles p1; p2 ;


; pm ; 1. We note from earlier discussions that lifting preserves the order of the system and the poles of the lifted system are raised individually by a power n. Thus the resulting lifted system has poles pn1 ; pn2 ;


; pnm ; 1 implying in nite gain for GL. For step type of reference signals, r() = 1 ,1  With the augmentation of a fast-rate integrator on the plant side, the signals after inverse lifting would be 4ui given by the closed-loop expression

4ui() = CL;i()e() =

CL;i r() n X 1 + CL;iGLI;i

(10)

i=1

Since GLI has in nite gain, using the nal-value theorem we can observe that the steady state error goes to zero at sampling instants. This means that 4ui (1) = CL;i (1)e(1) = 0, where 4ui is the input to the augmented plant GL I . This means that the fast-rate inputs to the plant are constant at steady-state. With a zero-order hold, the outputs should be steady.

2

The above argument shows that for set-point tracking with step type reference signals and plants with nite gains (Type 0), the controller gains have to be the same in order to avoid intersample oscillations. For plants with in nite gains i.e., the presence of an integrator in the plant, there is no restriction on the controller gains. Hence, for design purposes for stable plants, for step type reference signals, the lifted controller is built for the lifted augmented plant model. The lifted model can be either obtained by lifting the augmented plant model or the augmenting the original plant model with the lifted integrator LGI L,1, where GI is the fast-rate discrete integrator. For plants with integrators, Theorem 2 guarantees ripple-free steady-state output.

9

Controller with non−identical gains 0.2

Lifted signal

0.15

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100 Time

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Inverse lifted signal

0.2

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Figure 3: Inverse lifting of the lifted signal introduces oscillations in the inputs (n = 5)

3.3 Simulation results Here, we show the results obtained in the earlier discussion on (1) oscillations in continuous output and (2) providing a benchmark for a fair comparison of multirate and single-rate systems. We consider a SISO system given by 1 g^(s) = (10s + 1)(25 s + 1) with a base sampling period hb = n or hf = 1. We rst consider the issue of oscillations in the output of the plant when the gains of the lifted controller are non-identical. The closed-loop setup is similar to that shown in Fig.2. The inputs are sampled at the rate hf and the outputs are sampled at the rate hb . The SISO system is discretized at the fast-rate hf to get Gf and then the lifted system is obtained from equation (1). The continuous inputs and outputs are sampled at a relatively faster-rate than the fastest rate (i.e., almost continuous signals) in the system and we have designed an LQR controller for illustration purposes. Results are shown for a step change in the set-point. Figure 5 shows the oscillations in the output for non-identical controller gains and no oscillations with identical controller gains. The step change is given at t = 50 and Fig.5 is shown for later times. FIgure 3 shows how inverse lifting introduces oscillations in the inputs. THe lifted signal is the output of the lifted controller. The circled points on the multivariable lifted signal are the values at the consecutive instants for the inverse lifted signal. Thus dierent steady-states introduce oscillations in the fast-rate input signal due to inverse lifting. Input pro les for both the cases can be seen in Fig.6. Figure 4 shows the output and input pro les for the discretized plant augmented with a fast-rate integrator. Observe that the output tracks the step signal with no oscillations.

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Fast−rate plant augmented with fast−rate discrete integrato r 1.5

Output

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Figure 4: Output and input pro les for the plant augmented with integrator -set point tracking (n = 5)

With identical controller gains 0.28

Fast−rate output

0.275 0.27 0.265 0.26 0.255 0.25 100

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With unidentical controller gains

Fast−rate output

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Figure 5: Outputs of the closed-loop system with non-identical and identical controller gains -set point tracking (n = 5)

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Identical Controller gains Unidentical Controller gains

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Figure 6: Inputs to the plant for the closed-loop system with non-identical and identical controller gains -set point tracking (n = 5)

4 A continuous-time approach to the multirate control problem In the earlier section, we presented an application of lifting techniques to multirate control. Here, we present an approach based on continuous-time performance while using discrete lifting techniques. One of the primary reasons for builiding a multirate controller is to improve intersample ripples. Design based on the continuous-time performance of the closed-loop system can be a choice since it is impractical to measure the output at a faster rate, it is not possible to design with an objective function comprising fast-rate states/outputs. Hence it is motivating to derive an equivalent discretetime system given a continuous-time cost function. Results for single-rate systems are presented in (Chen and Francis, 1991b). We consider the design problem in the continuous-time domain in the LQR framework and derive the equivalent multirate system. Another motivating factor is that this type of design provides a fair benchmark for comparing the closed-loop performances of multirate and single-rate systems. In (Chen and Francis, 1991b), the authors discuss how to arrive at an equivalent discrete-system given a continuous-time cost function. This discussion holds good for a single-rate MIMO system. We extend the results here to multirate MIMO systems. Given a continuous-time state-space model G,

x_ = Ax + Bu z = C1x + D1 u y = C2x

(11) (12) (13)

where z is the ctitious signal whose k:k2 we want to minimize, we can arrive at the following equivalent single-rate discrete-system sampled with the base sampling period, hb (Chen and Francis, 12

z

z L

Gf Sn

y

u

u L -1

CL

Figure 7: Equivalent multirate system : Output sampled uniformly at a slower rate than the input 1991b). ~ k + Bu ~ k x_ k = Ax zk = C~1 xk + D~1 uk yk = C2 xk

(14) (15) (16)

where

A~ = eZAh h hA B~ = e Bdt b

b

b

0

and C~1 and D~1 are the solutions to the equation Z hb 0 0 ~ ~ ~ ~ [C1 D1 ] [C1 D1 ] = etA [C1 D1]0 [C1 D1 ] etAdt 0

wherein,

A = A0 B0

!

The integral on the right-hand side is arrived at by assuming a zero-order hold H with a period hb . It can be evaluated using Lemma 10.5.3 in (Chen and Francis, 1995). A discrete-controller can be designed either in the LQR framework or the H2 framework. For the LQR problem, with weighting matrices Qc and Rc in the continuous-time domain, Z1 Jc = (x0 Qcx + u0 Rcu)dt 0

we can obtain the state-space model given in equations (11)-(13) by noting that " 1=2 # " # 0 Q c C1 = 0 D1 = R1=2 c and then arrive at the equivalent cost function in the discrete domain 1 X Jd = (x0k Qd xk + u0k Rd uk + 2x0k Nd uk ) k=0

13

(17)

Qd , Rd and Nd are obtained from C~1 and D~1 as ! Qd Nd = [C~ D~ ]0[C~ D~ ] 1 1 1 1 Nd0 Rd If G~ f represents system described in equations (14)-(16) obtained at the fast-rate hb =n then for the multirate system, we can show using lifting techniques that the lifted system " ~ ,1 # ~ GL = SLLGGf~LL,1 n

f

represents the system from um to (z m ; ym ). Here, z m is the lifted ctitious signal obtained in terms of the available states and lifted inputs. Using lifting techniques, we can obtain a state-space model for LG~ f L,1 (Chen and Francis, 1995). Representing this model in transfer function form from um to z m as " ~ ~L1 # B A L 1 g~L = C~ D~ L1 L1 the equivalent QL , RL and NL in the cost-function for the lifted system 1 X (18) JL = (x0m QLxm + u0m RLum + 2x0m NL um) are obtained as

m=0

Q L NL NL0 RL

!

= [C~L1 D~ L1 ]0 [C~L1 D~ L1 ]

Note that the states are available at the same rate in both the multirate and the single-rate system (at the slow rate) while the inputs are available at the fast rate in the former case. It is observed that the weightings on the states in both the cost functions given in equations (17) and (18) are equal, while Rd and Nd are distributed in a particular way (given below) among the fast sampled inputs in the multirate system and the relation between the weightings for both the cost functions are stated below. The following theorem is stated for SISO systems. Theorem 3 Consider the continuous-time LQR cost function for a stable LTI continuous-time system G Z1 Jc = (x0 Qcx + u0 Rcu)dt 0 If Qd , Rd and Rd are the equivalent weights in the discrete-time cost function Jd for the single-rate system at the slow-rate, and QL , RL , NL are the equivalent weightings in the discrete-time cost function JL for the lifted slow-sampled fast-control multirate system with sampling ratios of the outputs and inputs as n, then the following relations hold good : Qd = QL (19) n n XX Rd = RLij (20)

Nd = i

i=1 j =1 n X NLij j =1

i = 1;


; ns

(21)

where RL = [RLij ] and NL = [NLij ] and ns is the order of the system G. Ndi is the weighting between the ith state and the input, Nd = [Nd1 Nd2


Ndns ]T

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Proof: Proof of this theorem is not included here for the sake of brevity. However, it is available from the authors. Extension to the multivariable case is under progress..

The approach given here provides a uniform basis for comparing the performance of single-rate and multirate systems. In practice, the achieved continuous-time performance can be computed using equations (17) and (18).

5 Conclusions and Future work Lifting techniques provide a suitable framework for the design of multirate controllers. Discrete lifting is shown to introduce non-uniform steady-state gains in the lifted models which lead to intersample ripples in the outputs of the closed-loop multirate system. For a slow-rate fast-control stable SISO system, we have shown here that for step changes in setpoints, if the lifted controller has identical gains then the outputs of the closed-loop multirate system are oscillation free during the intersample instants. Multirate controllers with conventional integrators do not ensure ripple free outputs while inclusion of a fast-rate integrator on the plant side ensures that outputs do not have intersample ripples. A continuous-time approach for multirate controller design is proposed in this work. It is indicated that the continuous-time design framework provides a fair benchmark for comparing the performance of single-rate and multirate systems. Our future goals include  Establishing the bene ts of multirate control over conventional control.  Incorporation of constraints in the multirate framework via model predictive control.  Extensions of the results in section 3 to MIMO systems.

References Araki, M. and K. Yamamoto (1986). Multivariable multirate sampled-data systems:state-space description, transfer characteristics and nyquist criterion. IEEE Transactions on Automatic Control, AC-31(2), 145{154. Chen, T. and B. Francis (1991a). Linear time-varying H2 -optimal control of sampled-data systems. Automatica, 27, 963{974. Chen, T. and B. Francis (1995). Optmal Sampled-Data Control Systems. Springer-Verlag. London. Chen, T. and Bruce Francis (1991b). H2 -optimal sampled data control. IEEE Transactions on Automatic Control 36(4), 387{397. Chen, T. and Li Qiu (1994). H1 design of general multirate sampled-data control systems. Automatica, 30(7), 1139{1152. Crochiere, R.E. and L.R. Rabiner (1983). Multirate Digital Signal Processing. Prentice-Hall Inc.. Eaglewoods Cli, New Jersey, USA. Franklin, G.F. and A. Emami-Nacini (1986). Design of ripple-free multivariable robust servomechanisms. IEEE Transactions on Automatic Control 31, 661{664. 15

Glasson, D.P. (1983). Development and applications of multirate digital control. IEEE Control Systems Magazine 3(4), 2{8. Gudi, R.D. and S.L. Shah (1993). The role of adaptive multirate Kalman lter as a software sensor and its application to a bioreactor. In: Proc. of 12th IFAC World Congress. Sydney, Australia. Kalman, R.E. (1960). A new approach to linear ltering and prediction problems. Trans. ASME Journal of Basic Engineering, 82, 34{45. Kalman, R.E. and J.E. Bertram (1959). A uni ed approach to the theory of sampling systems. J. Franklin Inst., 267, 405{436. Kranc, G.M (1957). Input-output analysis of multirate feedback systems. IRE Transactions on Automatic Control, 3, 21{28. Meyer, R.A. and C.S. Burrus (1990). A uni ed analysis of multirate and periodically time-varying digital lters. IEEE Transactions on Automatic Control, AC-35, 423{433. Ravi, R., O.O. Khargonekar, K.D. Minto and C.N. Nett (1990). Controller parametrization for time-varying multirate plants. IEEE Transactions on Automatic Control, AC-35, 1253{2162.

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