Least squares based self-tuning control of dual-rate systems

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2004; 18:697–714Published online in Wiley InterScience (www.interscience.wiley.com) DOI:10.1002/acs.828

Least squares based self-tuning control of dual-rate systems

Feng Ding1,2,z and Tongwen Chen2,n,y

1Department of Automation, Tsinghua University, Beijing 100084, People’s Republic of China2Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2V4

SUMMARY

A polynomial transformation technique is used to obtain a frequency-domain model for a dual-rate systemin which the output sampling period is an integer multiple of the input updating period. Based on thismodel, a self-tuning control algorithm is proposed by minimizing output tracking error criteria fromdirectly the dual-rate input–output data. Convergence properties of the algorithm are analysed in detail inthe stochastic framework. The output tracking error at the output sampling instants has the property ofminimum variance. It is shown that the control algorithm can achieve virtually optimal controlasymptotically, ensuring the closed-loop systems to be stable and globally convergent. A simulationexample illustrates the self-tuning scheme presented. Copyright # 2004 John Wiley & Sons, Ltd.

KEY WORDS: multirate systems; sampled-data systems; multirate modelling; self-tuning regulator;adaptive control; convergence properties; least squares

1. INTRODUCTION

In many industry applications, the outputs are sampled at slower rates than the controlupdating rates, mostly due to hardware limitation [1–3]. A typical example is the control of thebottom and top composition products of a distillation column by acting on the reflux and vaporflow rates; it is apparent that the control variables can be quickly manipulated, while infrequentand delayed composition measurements are obtained by gas chromatography [4]. In such casesseveral sample rates co-exist, it is necessary to configure a control system to achieve a desiredclosed-loop system performance. This paper is concerned with a dual-rate case where the outputsamples are at a relative slow rate, whereas the control signal is updated faster.

More generally, the study of multirate systems goes back to the early 1950s. The firstimportant work was performed by Kranc [5] on the switch decomposition technique. During the

Contract/grant sponsor: Natural Sciences and Engineering Research Council of Canada

Contract/grant Sponsor: National Natural Science Foundation of China

Accepted 22 June 2004Copyright # 2004 John Wiley & Sons, Ltd.

yE-mail: [email protected] a Research Associate at the University of Alberta, Edmonton, Canada.

nCorrespondence to: T. Chen, Department of Electrical and Computer Engineering, University of Alberta, Edmonton,Alberta, Canada T6G 2V4.

last decade, Al-Rahmani and Franklin [6, 7] studied multirate LQG/LQR optimal control; Chenand Qiu [8], Qiu and Chen [9, 10], and Sagf .oors et al. [11] studied H2 and H1 optimal control ofmultirate systems, considering the causality constraint. In the process control literature, Leeet al. [12], Scattolini and Schiavoni [13], Ling and Lim [14], and Sheng et al. [15] studied model-based predictive control of multirate systems; Scattolini [4], Albertos el al. [16], and Zhang et al.[17] investigated adaptive control involving dual-rate/multirate systems.

One motivation for the work in this paper is inferential control. In this area, Lee and Morari[18] developed a generalized inferential control scheme and discussed various optimal controlproblems for multirate/dual-rate systems; Li et al. [3, 19] applied dual-rate modelling to octanequality inferential control. However, most control algorithms reported in the area of multiratesystems assume that the parameters of multirate models are known, which is usually not thecase. Also, most theoretical results on parameter estimation-based adaptive control assume thatboth the estimator and the controller are updated at the same rate, e.g. the well-known (AAstr .oomand Wittenmark self-tuning regulator [20]. These results are not suitable for the dual-ratesetting. For dual-rate sampled-data control systems, we expect that the control law is updated ata fast rate even if the output is sampled at a relative slow rate.

In the field of dual-rate sampled-data adaptive control, the algorithm presented by Kanniahet al. [21] is based on a parameterized model with its AR coefficients corresponding to the fastsampling rate and the MA coefficients to the slow sampling rate. Since the prediction andcontrol are all based on the slow sampling rate, the desired fast-rate system performance maynot be achieved. Also, Zhang et al. [17] studied an indirect model reference multirate adaptivecontrol; Mitsuaki et al. [22] presented a least-squares-based self-tuning control algorithm. Butthese algorithms handle only noise-free systems. Scattolini [4] presented a gradient-basedadaptive control algorithm for multirate systems based on CARIMA models from lifted state-space models. In multirate stochastic systems with noise, to our best knowledge, the controlproblems based on model identification have not been fully investigated, especially the self-tuning control and its convergence properties based on multirate data directly, which are thefocus of this work.

The paper is organized as follows. In Section 2, we simply introduce the adaptive controlscheme of dual-rate systems. In Section 3, using a polynomial transformation technique, weestablish the mathematical model for dual-rate systems and a least-squares-based self-tuningcontrol. We prove the global convergence of the control algorithm proposed in Sections 4 and 5.In Section 6, we give an illustrative example demonstrating the effectiveness of the algorithmproposed in the paper. Finally, we offer some concluding remarks in Section 7.

2. PROBLEM FORMULATION

The focus of this paper is a class of multirate systems}the dual-rate systems}as depicted inFigure 1, where Pc is a continuous-time process; the input ucðtÞ to Pc is produced by a zero-order

Figure 1. The dual-rate system.

Copyright # 2004 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2004; 18:697–714

F. DING AND T. CHEN698

hold Hh with period h; processing a discrete-time signal uðkÞ; the output ycðtÞ of Pc is sampled bya sampler Sqh with period qh (q being a positive integer), yielding a discrete-time signal yðkqÞwith period qh: The input–output data available are

* fuðkÞ : k ¼ 0; 1; 2; . . .g at the fast rate, and* fyðkqÞ : k ¼ 0; 1; 2; . . .g at the slow rate.

Suppose that due to physical constraints, the intersample outputs, yðkqþ jÞ; j ¼ 1; 2; . . . ; q� 1;are not available, and thus we have missing output samples. We will refer to fuðkÞ; yðkqÞg as thedual-rate measurement data.

The adaptive control scheme we propose is shown in Figure 2, where yrðkÞ denotes adeterministic reference input or desired output signal, eðiqÞ a random noise with zero mean. Forsuch a scheme to work, we can exploit an identification algorithm to produce the estimates #yy ofthe unknown system parameters based on the dual-rate data fuðkÞ; yðkqÞg; and compute theintersample (missing) outputs by using the estimated model and input uðkÞ: In order to feed backto the controller a fast rate signal yf ðkÞ; representing the output yðkÞ; we use the slow sampledoutput yðiqÞ every q period, giving yð0Þ; yðqÞ; and yð2qÞ; etc., and use the estimated output#yyðiqþ jÞ to fill in the missing samples in yðkÞ: In Figure 2, yf ðkÞ connects to yðiqÞ at times k ¼ iq;and connects to #yyðiqþ jÞ at k ¼ iqþ j; j ¼ 1; 2; . . . ; ðq� 1Þ: Thus the output of the switch is afast rate signal given by yf ðkÞ: Due to the periodic switch, the fast rate signal yf ðkÞ can beexpressed as

yf ðkÞ ¼yðiqÞ; k ¼ iq

#yyðiqþ jÞ; k ¼ iqþ j; j ¼ 1; 2; . . . ; ðq� 1Þ

(ð1Þ

To summarize, the dual-rate adaptive control scheme uses a fast single-rate controller and aperiodic switch. It is conceptually simple, easy to implement in digital computers, and practicalfor industry.

The objective of this paper is to propose an algorithm to estimate the intersample outputsfyðkqþ jÞ : j ¼ 1; 2; . . . ; ðq� 1Þg based on the given dual-rate measurement data, design anadaptive controller so as to make the output yðkÞ track a given desired output yrðkÞ byminimizing the tracking error criterion function

J½uðkÞ� ¼ Ef½yf ðkþ dÞ � yrðkþ dÞ�2jFk�1g ð2Þ

Figure 2. The adaptive control scheme ð j ¼ 1; 2; . . . ; q� 1Þ:


SELF-TUNING CONTROL OF DUAL-RATE SYSTEMS 699

and study the properties of the closed-loop system. Here, d represents the system delay, fFkg isthe s algebra sequence generated by the observations up to and including time k:

3. MODELLING AND CONTROL ALGORITHM OF DUAL-RATE SYSTEMS

Setting the noise e to be zero in Figure 2, we assume that the open-loop transfer function fromuðkÞ to yðkÞ takes the following real-rational form:

P1ðzÞ ¼z�dbðzÞaðzÞ

or yðkÞ ¼z�dbðzÞaðzÞ

uðkÞ ð3Þ

with

aðzÞ ¼ 1þ a1z�1 þ a2z

�2 þ � � � þ anz�n

bðzÞ ¼ b0 þ b1z�1 þ b2z

�2 þ � � � þ bnz�n

Here, d denotes the system delay and z�1 represents a unit backward shift operator at the fastrate: z�1uðkÞ ¼ uðk� 1Þ:

This model in (3) is not suitable for dual-rate adaptive control because it would involve theunavailable outputs fyðkqþ jÞ : j ¼ 1; 2; . . . ; ðq� 1Þg: To obtain a model that we can usedirectly on the dual-rate data, by a polynomial transformation technique, P1ðzÞ needs to beconverted into a form so that the denominator is a polynomial in z�q instead of z�1:

For a general discussion, let the roots of aðzÞ be zi to get

aðzÞ ¼Yni¼1

ð1� ziz�1Þ

Define

fqðzÞ ¼Yni¼1

ð1þ ziz�1 þ z2i z

�2 þ � � � þ zq�1i z�qþ1Þ

Multiplying the numerator and denominator of P1ðzÞ by fqðzÞ transforms the denominator intothe desired form:

P2ðzÞ ¼z�dbðzÞfqðzÞ

aðzÞfqðzÞ¼:

z�dbðzÞaðzÞ

ð4Þ

or

aðzÞyðkÞ ¼ z�dbðzÞuðkÞ ð5Þ

with

aðzÞ ¼ 1þ a1z�q þ a2z�2q þ � � � þ anz�qn

bðzÞ ¼b0 þ b1z�1 þ b2z

�2 þ � � � þ bqnz�qn



Equation (4) is the frequency-domain model for the dual-rate system and it has the advantagethat the denominator is a polynomial of z�q; arising from here is a recursive equation using onlyslowly sampled outputs. The control algorithm we propose later for dual-rate systems will bebased on this model which does not involve the unavailable intersample outputs.

Next, we derive an adaptive control algorithm based on the model discussed in (5). Define theparameter vector y and information vector jðkÞ as

y ¼ ½a1 a2 � � � an b0 b1 � � � bqn�T 2 RN ; N :¼ qnþ nþ 1

jðk� dÞ ¼ ½�yðk� qÞ � yðk� 2qÞ � � � � yðk� qnÞ

uðk� dÞ uðk� d � 1Þ � � � uðk� d � qnÞ�T 2 RN

Notice that y contains all parameters in the model in (4) to be estimated, and jðk� dÞ uses onlyavailable dual-rate data}if k is an integer multiple of q; then jðk� dÞ contains only the pastmeasurement outputs (slow rate) and inputs (fast rate). Substituting the polynomials aðzÞ andbðzÞ into (5) leads to the following regression equation:

yðkÞ ¼ jTðk� dÞy ð6Þ

or

yðkþ dÞ ¼ jTðkÞy

Let yrðkÞ be a desired output signal, define the output tracking error

xðkþ dÞ ¼ yðkþ dÞ � yrðkþ dÞ

If the control signal uðkÞ is chosen according to the equation yrðkþ dÞ ¼ jTðkÞy; then thetracking error xðkþ dÞ approaches zero finally. For stochastic systems, based on the model in(6), introducing a disturbance term vðkÞ; we have

yðkÞ ¼ jTðk� dÞyþ vðkÞ ð7Þ

where fvðkÞg is assumed to be a zero-mean random white noise sequence. Let #yy be the estimateof unknown parameter vector y; then #yyðkþ dÞ ¼ jTðkÞ#yy is the best output prediction, which iscomputed by the intersample output estimator in Figure 2. Replacing y by #yy and minimizing thecriterion function in (2) yields the control law of the form:

yrðkþ dÞ ¼ jTðkÞ#yy ð8Þ

Replacing k in (7) by kq gives

yðkqÞ ¼ jTðkq� dÞyþ vðkqÞ ð9Þ

where

jðkq� dÞ ¼ ½�yðkq� qÞ � yðkq� 2qÞ � � � � yðkq� qnÞ

uðkq� dÞ uðkq� d � 1Þ � � � uðkq� d � qnÞ�T



Let #yyðkqÞ be the estimate of y at current time kq:We propose the self-tuning control algorithmfor the dual-rate system in (9) as follows:

#yyðkqÞ ¼ #yyðkq� qÞ þ PðkqÞjðkq� dÞ½yðkqÞ � jTðkq� dÞ#yyðkq� qÞ� ð10Þ

#yyðkqþ jÞ ¼ #yyðkqÞ; j ¼ 0; 1; . . . ; q� 1 ð11Þ

P�1ðkqÞ ¼ P�1ðkq� qÞ þ jðkq� dÞjTðkq� dÞ; Pð0Þ ¼ PTð0Þ > 0 ð12Þ

#yyðkqÞ ¼ ½#aa1ðkqÞ #aa2ðkqÞ � � � #aanðkqÞ #bb0ðkqÞ #bb1ðkqÞ � � � #bbqnðkqÞ�T ð13Þ

Based on (8), the control law is given by

jTðkqþ jÞ#yyðkqÞ ¼ yrðkqþ d þ jÞ; j ¼ 0; 1; . . . ; q� 1 ð14Þ

The control signal uðkqþ jÞ in (14) may be obtained from the following recursive equation:

uðkqþ jÞ ¼1

#bb0ðkqÞyrðkqþ d þ jÞ þ

Xni¼1

#aaiðkqÞyðkqþ d þ j � iqÞ

"

�Xnqi¼1

#bbiðkqÞuðkqþ j � iÞ

#; j ¼ 0; 1; . . . ; q� 1 ð15Þ

Here, a difficulty arises in that on the interval ½kq; kqþ qÞ; except for j ¼ q� d; the expressionon the right-hand side of (15) contains the future missing outputs yðkqþ j1Þ if j1 :¼ d þ j � iq >0; and the past missing outputs yðkq� j2Þ if j2 :¼ �d � j þ iq > 0 and j2 is not an integer multipleof q: In fact, only when j ¼ q� d; the control term uðkqþ jÞ does not involve the missingoutputs, and can be generated by

uðkqþ q� dÞ ¼1

#bb0ðkqÞyrðkqþ qÞ þ

Xni¼1

#aaiðkqÞyðkqþ q� iqÞ

"

�Xnqi¼1

#bbiðkqÞuðkqþ q� d � iÞ

#ð16Þ

So it is impossible to compute the control law by (15) and to realize the algorithm in (10)–(15).Our solution is based on the adaptive control scheme stated in Section 2: These unknownoutputs yðkqþ jÞ in (15) are replaced by their estimates #yyðkqþ jÞ: Hence,

uðkqþ jÞ ¼1

#bb0ðkqÞyrðkqþ d þ jÞ þ

Xni¼1

#aaiðkqÞ #yyðkqþ d þ j � iqÞ

"

�Xnqi¼1

#bbiðkqÞuðkqþ j � iÞ

#; j ¼ 0; 1; . . . ; q� 1; j=q� d ð17Þ



To initialize the control algorithm in (10)–(13), (16) and (17), we take Pð0Þ ¼ p0I with p0normally a large positive number and I an identity matrix of appropriate dimension, and#yyð0Þ ¼ #yy0; some small real vector. Notice that the parameter estimate #yy is updated every q (fast)samples, namely, at the slow rate; so is the covariance matrix P; in between the slow samples, wesimply hold #yy unchanged. Thus, every time #yy is updated, we have q new input samples and onenew output sample.

It is easy to see that by defining

LðkqÞ :¼PðkqÞjðkq� dÞ

¼Pðkq� qÞjðkq� dÞ

1þ jTðkq� dÞPðkq� qÞjðkq� dÞ

the covariance matrix P can be updated as follows:

PðkqÞ ¼Pðkq� qÞ �Pðkq� qÞjðkq� dÞjTðkq� dÞPðkq� qÞ

1þ jTðkq� dÞPðkq� qÞjðkq� dÞ

¼ ½I � LðkqÞjTðkq� dÞ�Pðkq� qÞ

When q ¼ 1; the control algorithm in (10)–(14) reduces to the well-known (AAstr .oom–Wittenmark self-tuning regulator.

4. THE OUTPUT TRACKING PERFORMANCE

We assume that fvðkÞ;Fkg is a martingale difference sequence defined on a probability spacefO;F;Pg; where fFkg is the s algebra sequence generated by fvðkÞg; i.e. Fk ¼ sðvðkÞ;vðk� 1Þ; vðk� 2Þ; . . .Þ; and that the noise sequence fvðkÞg satisfies the following conditions [23]:

(A1) E½vðkÞjFk�1� ¼ 0; a.s.;(A2) E½v2ðkÞjFk�1� ¼ s2vðkÞ4 %ss2v51; a.s.;(A3) lim supk!1 ð1=kÞ

Pki¼1 v2ðiÞ4 %ss2v51; a.s.

That is, fvðkÞg is an independent random noise sequence with zero mean and bounded time-varying variance.

Let us introduce some notation first. Let X be a square matrix; the symbols lmax½X � representsthe maximum eigenvalue of X ; respectively; li½X � represent the ith eigenvalue of X : f ðkÞ ¼oðgðkÞÞ represents f ðkÞ=gðkÞ ! 0 as k ! 1; for gðkÞ50; we write f ðkÞ ¼ OðgðkÞÞ if there exists aconstant dm > 0 such that jf ðkÞj4dmgðkÞ:

Define

rðkqÞ ¼ tr½P�1ðkqÞ�

It follows easily that

rðkqÞ ¼N

p0þ

Xki¼1

jjjðiq� dÞjj2



From here we get

rðkqÞ ¼ rðkq� qÞ þ jjjðkq� dÞjj2

rðkqÞ ¼ l1½P�1ðkqÞ� þ l2½P�1ðkqÞ� þ � � � þ lN ½P�1ðkqÞ�

4Nlmax½P�1ðkqÞ�

jP�1ðkqÞj ¼ l1½P�1ðkqÞ�l2½P�1ðkqÞ� � � � lN ½P�1ðkqÞ�

4lNmax½P�1ðkqÞ�

4 rNðkqÞ

and

ln jP�1ðkqÞj4N ln rðkqÞ

or

ln jP�1ðkqÞj ¼ Oðln rðkqÞÞ ð18Þ

In order to study the output tracking performance of the algorithm, the following lemma isrequired.

Lemma 1

The following inequality holds:X1i¼1

jTðiq� dÞPðiqÞjðiq� dÞfln jP�1ðiqÞjgc

51 a:s:; for any c > 1

where j � j is the matrix determinant. (If jP�1ð0Þj is too small, we then begin the summation atsome i ¼ i0 > 1 instead of i ¼ 1:)

Proof

From the definition of PðkqÞ; we have

P�1ðiq� qÞ ¼P�1ðiqÞ � jðiq� dÞjTðiq� dÞ

¼P�1ðiqÞ½I � PðiqÞjðiq� dÞjTðiq� dÞ�

Taking determinants on both sides gives

jP�1ðiq� qÞj ¼ jP�1ðkqÞjjI � PðkqÞjðkq� dÞjTðkq� dÞj

¼ jP�1ðiqÞj½1� jTðiq� dÞPðiqÞjðiq� dÞ�

Thus

jTðiq� dÞPðiqÞjðiq� dÞ ¼jP�1ðiqÞj � jP�1ðiq� qÞj

jP�1ðiqÞjð19Þ



Noting that jP�1ðiqÞj is a non-decreasing function of i; dividing both sides of (19) byfln jP�1ðkqÞgc gives

X1i¼1

jP�1ðiqÞj � jP�1ðiq� qÞjjP�1ðiqÞjfln jP�1ðiqÞjgc

¼X1i¼1

Z jP�1ðiqÞj

jP�1ðiq�qÞj

dx

jP�1ðiqÞjfln jP�1ðiqÞjgc

4X1i¼1

Z jP�1ðiqÞj

jP�1ðiq�qÞj

dx

xðln xÞc

¼Z jP�1ð1Þj

jP�1ð0Þj

dx

xðln xÞc

¼�1

c� 1�

1

fln xgc�1

jP�1ð1ÞjjP�1ð0Þj

��¼

1

c� 1

1

fln jP�1ð0Þjgc�1�

1

fln jP�1ð1Þjgc�1

� �

51 a:s:; c > 1 &

We shall prove the main results of this paper by formulating a martingale process and byusing stochastic process theory and the martingale convergence theorem [23, Lemma D.5.3].

Theorem 1

For the dual-rate system in (9), assume that (A1)–(A3) hold, bðzÞ is stable, the system delay d4qis known, and the reference input yrðkÞ is bounded in the sense

(A4) jyrðkÞj51:Then the adaptive control algorithm in (10)–(13), (16) and (17) ensures that the output trackingerror at the output sampling instants has the property of minimum variance, i.e.

(1) limk!11k

Pki¼1 ½yrðiqÞ � yðiqÞ þ vðiqÞ�2 ¼ 0 a.s.;

(2) lim supk!11k

Pki¼1 Ef½yf ðiqÞ � yrðiqÞ�2jFiq�1g4 %ss2v51 a.s.

Proof

Define the parameter estimation error vector as

*yyðkqÞ ¼ #yyðkqÞ � y

Using (9) and (10) we have

*yyðkqÞ ¼ *yyðkq� qÞ þ PðkqÞjðkq� dÞ½jTðkq� dÞyþ vðkqÞ � jTðkq� dÞ#yyðkq� qÞ�

¼ *yyðkq� qÞ þ PðkqÞjðkq� dÞ½� *yyðkqÞ þ vðkqÞ� ð20Þ



where

*yyðkqÞ ¼ jTðkq� dÞ*yyðkq� qÞ ¼ jTðkq� dÞ#yyðkq� qÞ � jTðkq� dÞy ð21Þ

By using (9) and (14), for d4q; it follows that,

*yyðkqÞ ¼ yrðkqÞ � yðkqÞ þ vðkqÞ

Define a non-negative definite function

VðkqÞ ¼ *yyTðkqÞP�1ðkqÞ*yyðkqÞ

Using (9), (20) and (21), we have

VðkqÞ ¼ *yyTðkq� qÞP�1ðkqÞ*yyðkq� qÞ þ 2*yyTðkq� qÞjðkq� dÞ½� *yyðkqÞ þ vðkqÞ�

þ jTðkq� dÞPðkqÞjðkq� dÞ½� *yyðkqÞ þ vðkqÞ�2

¼ *yyTðkq� qÞ½P�1ðkq� qÞ þ jTðkq� dÞjðkq� dÞ�*yyðkq� qÞ

þ 2 *yyðkqÞ½� *yyðkqÞ þ vðkqÞ� þ jTðkq� dÞPðkqÞjðkq� dÞ½� *yyðkqÞ þ vðkqÞ�2

¼Vðkq� qÞ � ½1� jTðkq� dÞPðkqÞjðkq� dÞ� *yy2ðkqÞ

þ jTðkq� dÞPðkqÞjðkq� dÞv2ðkqÞ

þ 2½1� jTðkq� dÞPðkqÞjðkq� dÞ� *yyðkqÞvðkqÞ ð22Þ

Since *yyðkqÞ; jTðkq� dÞPðkqÞjðkq� dÞ are independent of vðkqÞ and Fkq�1-measurable, takingthe conditional expectation on both sides of (22) with respect to Fkq�1 and using (A1)–(A2)gives,

E½VðkqÞjFkq�1�4Vðkq� qÞ � ½1� jTðkq� dÞPðkqÞjðkq� dÞ� *yy2ðkqÞ

þ 2jTðkq� dÞPðkqÞjðkq� dÞ %ss2v

Let

WðkqÞ ¼VðkqÞ

½ln jP�1ðkqÞj�c; c > 1

Noting that ln jP�1ðkqÞj is non-decreasing, we have

E½WðkqÞjFkq�1�4Vðkq� qÞ

½ln jP�1ðkqÞj�c�

1� jTðkq� dÞPðkqÞjðkq� dÞ½ln jP�1ðkqÞj�c

*yy2ðkqÞ

þ2jTðkq� dÞPðkqÞjðkq� dÞ

½ln jP�1ðkqÞj�c%ss2v

4Wðkq� qÞ �1� jTðkq� dÞPðkqÞjðkq� dÞ

½ln jP�1ðkqÞj�c*yy2ðkqÞ

þ2jTðkq� dÞPðkqÞjðkq� dÞ

½ln jP�1ðkqÞj�c%ss2v ð23Þ



By using Lemma 1, it is clear that the sum of the last right-hand term of (23) for k from 1 to1 isfinite. Since

1� jTðkq� dÞPðkqÞjðkq� dÞ ¼ ½1þ jTðkq� dÞPðkq� qÞjðkq� dÞ��150

applying the martingale convergence theorem [23, Lemma D.5.3] to (23), we conclude thatWðkqÞ converges a.s. to a finite random variable, say, W0; i.e.

WðkqÞ ¼VðkqÞ

½ln jP�1ðkqÞj�c! W051 a:s:; or VðkqÞ ¼ Oð½ln jP�1ðkqÞj�cÞ a:s:

and

X1k¼1

1� jTðkq� dÞPðkqÞjðkq� dÞ½ln jP�1ðkqÞj�c

*yy2ðkqÞ51 a:s:

By using (18), it follows that,

X1k¼1

1� jTðkq� dÞPðkqÞjðkq� dÞ½ln rðkqÞ�c

*yy2ðkqÞ51 a:s:

Since jTðkq� dÞPðkqÞjðkq� dÞ ¼ oð1Þ; we have

X1i¼1

*yy2ðiqÞ½ln rðiqÞ�c

51 a:s: ð24Þ

As rðkqÞ ! 1; using the Kronecker lemma [23, Lemma D.5.5] yields

limk!1

1

½ln rðkqÞ�cXki¼1

*yy2ðiqÞ ¼ 0 a:s:

Since ½ln rðkqÞ�c ¼ oðrðkqÞÞ; we have

limk!1

k

rðkqÞ1

k

Xki¼1

*yy2ðiqÞ ¼ 0 a:s: ð25Þ

Since bðzÞ is stable, applying Lemma B.3.3 in Reference [23] to (9) and using (A3) yield

1

k

Xki¼1

u2ðiqÞ4c1

k

Xki¼1

y2ðiqÞ þ c2 a:s:

where ci represent finite positive constants. According to the definitions of rðkqÞ and jðkqÞ; it isnot difficult to get

rðkqÞk

4c3

k

Xki¼1

y2ðiqÞ þ c4

¼c3

k

Xki¼1

½yrðiqÞ � *yyðiqÞ þ vðiqÞ�2 þ c4

4c5

k

Xki¼1

*yy2ðiqÞ þ c6 a:s:



Thus, from (25)

0 ¼ limk!1

ð1=kÞPk

i¼1 *yy2ðiqÞrðkqÞ=k

5 limk!1

ð1=kÞPk

i¼1 *yy2ðiqÞ

ðc5=kÞPk

i¼1 *yy2ðiqÞ þ c650 a:s:

and hence

limk!1

1

k

Xki¼1

*yy2ðiqÞ ¼ 0 a:s:

or

limk!1

1

k

Xki¼1

½yrðiqÞ � yðiqÞ þ vðiqÞ�2 ¼ 0 a:s: ð26Þ

Since

Ef½yrðkqÞ � yðkqÞ þ vðkqÞ�2jFkq�1g ¼E½ðyrðkqÞ � yðkqÞÞ2 þ 2yrðkqÞvðkqÞ

� 2yðkqÞvðkqÞ þ v2ðkqÞjFkq�1�

¼E½ðyrðkqÞ � yðkqÞÞ2jFkq�1� þ 0� 2s2vðkqÞ þ s2vðkqÞ

¼E½ðyrðkqÞ � yðkqÞÞ2jFkq�1� � s2vðkqÞ a:s:

and yf ðkqÞ ¼ yðkqÞ at the output sampling instants, we have

lim supk!1

1

k

Xki¼1

Ef½yf ðiqÞ � yrðiqÞ�2jFiq�1g ¼ lim supk!1

1

k

Xki¼1

Ef½yðiqÞ � yrðiqÞ�2jFiq�1g

¼ lim supk!1

1

k

Xki¼1

s2vðiqÞ4 %ss2v a:s:

This proves Theorem 1. &

Since single-rate systems belong to a special class of dual-rate systems with q ¼ 1; the resultsin Theorems 1 still hold for single-rate systems.

5. THE MISSING OUTPUT ESTIMATION

From (1) and (9), we have

yf ðkqÞ ¼ yðkqÞ ¼ jTðkq� dÞyþ vðkqÞ ð27Þ

yf ðkqþ jÞ ¼ #yyðkqþ jÞ; j ¼ 1; 2; . . . ; q� 1 ð28Þ

From Figure 2 and (5), all missing output estimates yf ðkqþ jÞ can be obtained by

#yyðkqþ jÞ ¼z�d #bbðkq; zÞ#aaðkq; zÞ

uðkqþ jÞ ð29Þ



where

#aaðkq; zÞ ¼ 1þ #aa1ðkqÞz�q þ #aa2ðkqÞz�2q þ � � � þ #aanðkqÞz�qn

#bbðkq; zÞ ¼ #bb0ðkqÞ þ #bb1ðkqÞz�1 þ #bb2ðkqÞz�2 þ � � � þ #bbqnðkqÞz�qn

#yyðkqÞ ¼ ½#aa1ðkqÞ #aa2ðkqÞ � � � #aanðkqÞ #bb0ðkqÞ #bb1ðkqÞ � � � #bbqnðkqÞ�

The output estimates #yyðkqþ jÞ can also be computed from the recursive equation:

#yyðkqþ jÞ þXni¼1

#aaiðkqÞ #yyðkqþ j � iqÞ ¼Xnqi¼0

#bbiðkqÞuðkq� d þ j � iÞ; j ¼ 0; 1; . . . ; q� 1 ð30Þ

This means

#yyðkqþ jÞ ¼ #jjTðkqþ jÞ#yyðkqÞ; j ¼ 1; 2; . . . ; q� 1

where

#jjðkqþ jÞ ¼ ½� #yyðkq� qþ jÞ � #yyðkq� 2qþ jÞ � � � � #yyðkq� qnþ jÞ

� uðkq� d þ jÞ uðkq� d þ j � 1Þ � � � uðkq� d þ j � qnÞ�T

Comparing (17) with (30), we find that the missing intersample output estimates #yyðkqþ jÞ; j ¼1; 2; . . . ; q� 1; equal the desired outputs yrðkqþ jÞ; so we have

yrðkqþ jÞ ¼ #yyðkqþ jÞ ¼ #jjTðkqþ jÞ#yyðkqÞ; j ¼ 1; 2; . . . ; q� 1 ð31Þ

#jjðkqþ jÞ ¼ ½�yrðkq� qþ jÞ � yrðkq� 2qþ jÞ � � � � yrðkq� qnþ jÞ

uðkq� d þ jÞ uðkq� d þ j � 1Þ � � � uðkq� d þ j � qnÞ�T

It is easy to understand that the unknown intersample outputs yðkqþ jÞ are replaced by thedesired outputs yrðkqþ jÞ in that our goal is to make yðkÞ track yrðkÞ: Hence, combining (16)with (31) generates the control signal sequence fuðkqþ jÞ; j ¼ 0; 1; . . . ; q� 1g based on theparameter estimates #yyðkqÞ obtained. Thus, the following theorem is easily established.

Theorem 2

Assume that the conditions of Theorem 1 hold, the open-loop system bðzÞ=aðzÞ is minimumphase. Then the adaptive control algorithm in (10)–(13), (16) and (17) ensures the closed-loopsystem to be stable and globally convergent with probability 1; moreover,

* The input and output variables are uniformly bounded, i.e.

lim supk!1

1

k

Xki¼1

½u2ðiÞ þ y2ðiÞ þ y2f ðiÞ�51 a:s:

* The average tracking error is less than %ss2v=q; i.e.

lim supk!1

1

k

Xki¼1

Ef½yf ðiÞ � yrðiÞ�2jFi�1g4%ss2vq

a:s:



Proof

Since yrðkÞ is bounded, from Theorem 1 and Condition (A3), we can draw that outputs yðkqÞ atthe output sampling instants are uniformly bounded, i.e.

lim supk!

1

k

Xki¼1

y2ðiqÞ4dy51 a:s:

Also, the intersample output estimates #yyðkqþ jÞ; j ¼ 1; 2; . . . ; ðq� 1Þ; satisfy

#yyðkqþ jÞ ¼ yrðkqþ jÞ; j ¼ 1; 2; . . . ; q� 1

So yf ðkqþ jÞ is bounded. According to (27) and (28), yf ðkÞ is bounded. Since bðzÞ=aðzÞ isminimum phase, so are bðzÞ=aðzÞ and aðzÞ=bðzÞ: uðkÞ is bounded in terms of Lemma B.3.3 in [23].Hence, we have

lim supk!1

1

k

Xki¼1

u2ðiÞ51 a:s: lim supk!1

1

k

Xki¼1

y2ðiÞ51 a:s:

lim supk!1

1

k

Xki¼1

y2f ðiÞ51 a:s:

That is, the input and output variables are uniformly bounded. Also,

lim supk!1

1

k

Xki¼1

Ef½yf ðiÞ � yrðiÞ�2jFi�1g

¼ lim supk!1

1

kq

Xki¼1

Ef½yðiqÞ � yrðiqÞ�2jFiq�1g

þ lim supk!1

1

kq

Xq�1

j¼1

Xki¼1

Ef½ #yyðiqþ jÞ � yrðiqþ jÞ�2jFiqþj�1g

Since yf ðiÞ ¼ #yyðiÞ ¼ yrðiÞ at the missing outputs, the second term on the right-hand side is zero,and the first term is less than %ss2v=q: Thus we complete the proof of Theorem 2. &

Theorems 1 and 2 show that the proposed control scheme in the dual-rate setting canachieve the property of minimum variance at the output sampling instants, just likethe (AAstr .oom–Wittenmark STR. Between the output sampling instants, we have#yyf ðkqþ jÞ ¼ #yyðkqþ jÞ ¼yrðkqþ jÞ; j ¼ 1; 2; . . . ; q� 1; which implies zero tracking error forintersamples.

In order to avoid generating uðkÞ with too large magnitudes, for a given small positive e; ifj #bb0ðkqÞj5e; we take #bb0ðkqÞ ¼ sgn½ #bb0ðkqÞ�e; where the sign function is defined by

sgnðxÞ ¼1; x50

0; x50

(



6. EXAMPLE

Assume that the discrete system model with period h ¼ 2 s takes the following form:

P1ðzÞ ¼z�dbðzÞaðzÞ

¼0:412z�1 þ 0:309z�2

1� 1:60z�1 þ 0:80z�2; d ¼ 1

Take qh ¼ 4 s and qh ¼ 6 s; and hence q ¼ 2 and 3; respectively. The model with a noise isgiven by

fqðzÞaðzÞyðkÞ ¼ z�dfqðzÞbðzÞuðkÞ þ vðkÞ

Use fvðkÞg as a white noise sequence with zero mean and variance %ss2v ; and take the desiredoutput to be

yrð400i þ jÞ ¼ ð�1Þi; i ¼ 0; 1; 2; . . . ; j ¼ 1; 2; . . . ; 400

We apply the adaptive control algorithm in Section 3 to this system, and the results for q ¼1; 2; 3 are shown in Figures 3–5, where yðkÞ represents the system output, yrðkÞ the desiredoutput. Figure 3 with q ¼ 1 is the simulated results of the (AAstr .oom–Wittenmark self-tuningregulator (STR).

In order to evaluate the accuracy for output tracking, we define an accumulative outputtracking error at the output sampling instants as

deðqÞ ¼XL0þLe�1

k¼L0

½yðkqÞ � yrðkqÞ�2

where L0 ¼ 160 and Le ¼ 3000 (data length). The quantity #ss2y ¼ deðqÞ=Le can be regarded as theestimated output tracking variance. For %ss2v ¼ 0:012 and %ss2v ¼ 0:102; the output tracking errorsdeðqÞ and #ss2y when varying q are shown in Table I. From Table I, we can see that the controlalgorithm proposed ðq ¼ 2; 3Þ and the (AAstr .oom–Wittenmark self-tuning regulator ðq ¼ 1Þ havealmost equal output tracking variance, which approximately equals to the noise variance %ss2v ;

500 1000 1500 2000 2500 3000-1.5

-1

-0.5

0

0.5

1

1.5

k

y(k)

, y r

(k)

y(k) y(k)y

r(k)

Figure 3. yðkÞ and yrðkÞ versus k ðq ¼ 1Þ of the (AAstr .oom–Wittenmark STR ð %ss2v ¼ 0:102Þ:



500 1000 1500 2000 2500 3000-1.5

-1

-0.5

0

0.5

1

1.5

k

y(k)

, y r(k

)

y(k) y(k)yr (k)

Figure 4. yðkÞ and yrðkÞ versus k ð %ss2v ¼ 0:102; q ¼ 2Þ:

500 1000 1500 2000 2500 3000-1.5

-1

-0.5

0

0.5

1

1.5

k

y(k)

, y r

(k)

y(k) y(k)yr (k)

Figure 5. yðkÞ and yrðkÞ versus k ð %ss2v ¼ 0:102; q ¼ 3Þ:

Table I. Output tracking errors under different noise variances.

%ss2v ¼ 0:012 %ss2v ¼ 0:102

q deðqÞ #ss2y deðqÞ #ss2y

1 0.30234 0:010042 30.20359 0:100342

2 0.30056 0:010012 29.83303 0:099722

3 0.29142 0:009862 29.13935 0:098562



that is, they all have the property of minimum variance at the output sampling instants. Also,decreasing the noise variance can improve the accuracy of the output tracking. These confirmthe results reported earlier.

7. CONCLUSIONS

A least-squares-based control algorithm for dual-rate systems is presented; the algorithm usesonly slow-rate output measurement data and generates a relative fast-rate control signal.Convergence performance of the proposed algorithm is analysed in detail in the stochasticframework. The algorithm achieves the desired control performance under certain conditions.The algorithm is also applied to a simulated system successfully, and the simulated results verifythe theoretical findings. The control method for the case d > q is currently being studied in thestochastic framework.

ACKNOWLEDGEMENTS

This research was supported by the Natural Sciences and Engineering Research Council of Canada and theNational Natural Science Foundation of China.

REFERENCES

1. Ohshima M, Hashimoto I, Takeda M, Yoneyama T, Goto F. Multirate multivariable model predictive control andits application to a semi-commercial polymerization reactor. Proceedings of the 1992 American Control Conference,Chicago, IL, 1992; 1576–1581.

2. Gudi RD, Shah SL, Gray MR. Multirate state and parameter estimation in an antibiotic fermentation with delayedmeasurements. Biotechnology and Bioengineering 1994; 44(11):1271–1278.

3. Li D, Shah SL, Chen T, Qi KZ. Application of dual-rate modeling to CCR octane quality inferential control. IEEETransactions on Control Systems Technology 2003; 11(1):43–51.

4. Scattolini R. Self-tuning control of systems with infrequent and delayed output sampling. IEE Proceedings, Part D,Control Theory and Applications 1988; 135(4):213–221.

5. Kranc GM. Input-output analysis of multirate feedback systems. IRE Transactions on Automatic Control 1957;3(1):21–28.

6. Al-Rahmani HM, Franklin GF. A new optimal multirate control of linear periodic and time-invariant systems.IEEE Transactions on Automatic Control 1990; 35(4):406–415.

7. Al-Rahmani HM, Franklin GF. Multirate control: a new approach. Automatica 1992; 28(1):35–44.8. Chen T, Qiu L. H1 design of general multirate sampled-data control systems. Automatica 1994; 30(7):1139–1152.9. Qiu L, Chen T. H2-optimal design of multirate sampled-data systems. IEEE Transactions on Automatic Control

1994; 39(12):2506–2511.10. Qiu L, Chen T. Multirate sampled-data systems: all H1 suboptimal controllers and the minimum entropy

controller. IEEE Transactions on Automatic Control 1999; 44(3):537–550.11. Sagf .oors MF, Toivonen HT, Lennartson B. State-space solution to the periodic multirate H1 control problem: a

lifting approach. IEEE Transactions on Automatic Control 2000; 45(12):2345–2350.12. Lee JH, Gelormino MS, Morari M. Model predictive control of multirate sampling-data systems: a state-space

approach. International Journal of Control 1992; 55(1):153–191.13. Scattolini R, Schiavoni N. A multirate model-based predictive controller. IEEE Transactions on Automatic Control

1995; 40(6):1093–1097.14. Ling KV, Lim KW. A state-space GPC with extensions to multirate control. Automatica 1996; 32(7):1067–1071.15. Sheng J, Chen T, Shah SL. Generalized predictive control for non-uniformly sampled systems. Journal of Process

Control 2002; 12(8):875–885.16. Albertos P, Salt J, Tormero J. Dual-rate adaptive control. Automatica 1996; 32(7):1027–1030.17. Zhang C, Middleton RH, Evans RJ. An algorithm for multirate sampling adaptive control. IEEE Transactions on

Automatic Control 1989; 34(7):792–795.



18. Lee JH, Morari M. Robust inferential control of multirate sampling-data systems. Chemical Engineering Science1992; 47(4):865–885.

19. Li D, Shah SL, Chen T. Analysis of dual-rate inferential control systems. Automatica 2002; 38(6):1053–1059.20. (AAstr .oom KJ, Wittenmak B. On self-tuning regulators. Automatica 1973; 9(2):185–199.21. Kanniah J, Malik OP, Hope GS. Self-tuning regulator based on dual-rate sampling. IEEE Transactions on

Automatic Control 1984; 29(8):755–759.22. Mitsuaki I, Masaki K, Hiroaki N. Ripple-suppressed multirate adaptive control. 15th Triennial World Congress of

the International Federation of Automatic Control, Barcelona, Spain, July 2002; 21–26.23. Goodwin GC, Sin KS. Adaptive Filtering Prediction and Control. Prentice-Hall: Englewood Cliffs, NJ, 1984.



Least squares based self-tuning control of dual-rate systems

Documents

Transcript of Least squares based self-tuning control of dual-rate systems