Switching Control using Generalized Sampled-Data Hold Functions

Shauheen Zahirazami and Amir G. Aghdam

Abstract— In this paper, switching control of linear time-invariant systems using generalized sampled-data hold func-tions is investigated. It is assumed that the plant model belongsto a finite set of known plants. The output of the system isperiodically sampled and a control signal is being generatedby means of a suitable hold function which solves the robustservomechanism problem for a family of plant models. It isalso desirable to achieve a digital control law that reduces thecomplexity of online computations.

I. INTRODUCTION

Servomechanism problem, known also as the output reg-ulation problem, plays a crucial role in control theory. Itconcerns with the design of a feedback control law forthe system, to asymptotically track the reference inputsand reject disturbances while maintaining the closed-loopstability.

Switching techniques have been the center of focus ofmany researchers in the past decade, specially when classicaladaptive control methods fail to stabilize the systems dueto, for instance, sudden changes in the dynamics of themodel. The earliest works to weaken the classical a prioriinformation required in adaptive methods can be tracked backto Morse and Martensson’s works [1], [2]. Switching of afamily of plant models {Pi : i ∈ p = {1,2, . . . , p}} using highperformance LTI controllers was first introduced by Millerand Davison in [3], [4] and [5], where it is assumed thateach plant-controller combination (Pi,K j) is stable iff i = j,i, j ∈ p, and that a bound on the unmeasurable disturbancesignal is given as a priori information. It was shown that thesystem would not switch to each controller more than once.

The idea of using generalized sampled-data hold functions(GSHF) instead of a simple zero-order hold (or first-orderhold) in control systems, was first introduced by Chammasand Leondes [6], and received a lot of attention [7], [8].Kabamba presented several advantages of using GSHFs incontrol systems, and proved that by using GSHF, one canobtain many of the advantages of state feedback controllers,without the requirement of using state estimation procedures;in particular, it was shown that GSHFs can significantlyimprove the performance of a closed-loop system. A method

This work has been supported by the Natural Sciences and EngineeringResearch Council of Canada under grant RGPIN-262127-03.

Shauheen Zahirazami is with the Department of Electrical and ComputerEngineering, Concordia University, Montreal, Quebec, Canada H3G 1M8shauheen@ieee.org

Amir G. Aghdam is with the Department of Electrical and ComputerEngineering, Concordia University, Montreal, Quebec, Canada H3G 1M8aghdam@ece.concordia.ca

was presented in [9] to design GSHFs, based on the dynam-ics of the system. It can achieve closed loop stability bysampling the output of the system and constructing the inputsignal as a function of the output samples in each interval.

It was shown in [10] that in many cases, digital controllerswith GSHFs can significantly improve the overall perfor-mance of a certain class of decentralized control system; alsoin [11] it was shown that the GSHFs can be used to modifythe structure of the digraph of a system in the discrete-timedomain, by removing certain interconnections in the discretetime equivalent model of the system.

In this paper, GSHFs are employed in the switchingmechanism instead of LTI controllers used in traditionalswitching control. The switching scheme used here is thediscrete time version of the approach presented in [3]. It isassumed that a GSHF denoted by fi, is designed for eachplant model Pi, in the family of plant models,

Π := {Pi : i ∈ p = {1,2, . . . , p}} (1)

and that the system switches between different GSHFs inproper time instants, until it finds the correct GSHF to controlthe system. It is also assumed that each plant-GSHF pair(Pi, f j) gives a stable equivalent discrete time model iff i = jand that a bound on the disturbance signal is given.

One of the objectives of this work is to propose a switchingcontroller which is computationally efficient in obtainingbounding functions or auxiliary signals compared to theexisting methods [3],[12]. The required control computa-tions are also much less than continuous-time switchingcontrol techniques. Moreover, the proposed switching controlmethod utilizes the benefits of output control using GSHFswhich can, in general, outperform traditional LTI controllers.On the other it is known that discrete-time controllers arevery effective in decentralized systems. Thus, the proposedmethod can be applied in a decentralized manner to achievebetter performance.

Notation: Throughout this paper, sample of a continuous-time signal z(t) at t = kT will be denoted by z[k], also thenorm of x ∈ R

n which we denote by ‖x‖ is the Holder2-norm; with A ∈ R

n×m, ‖A‖ denotes the correspondinginduced norm of A.

II. PROBLEM FORMULATION

Consider a strictly proper, controllable and observable, LTIsystem Pi(t), in a given finite set of plant models Π definedin (1).

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

TC4.4

0-7803-9354-6/05/$20.00 ©2005 IEEE 1081

∀t : Pi(t) ∈ Π (2)

The plant model is defined by the following state-spaceformulation:

{x(t) = Aix(t)+Biu(t)+Eiω(t)y(t) = Cix(t)+Fiω(t) (3a)

e = yre f − y (3b)

where x(t) ∈ Rni , i ∈ p is the state, u(t) ∈ Rm is the controlinput, y(t) ∈ Rr is the output, ω(t) ∈ Rv is the disturbancesignal and e(t)∈Rr is the error signal, which is the differencebetween the output and the reference signals. For simplicityand without loss of generality it will be assumed that m = r.

Figure 1 shows the closed loop model of the system.

Fig. 1. Closed-loop model with GSHF

The control signal is constructed by multiplying the discretesamples of the input and the GSHF, i.e.

u(t) = fi(t).u[k]fi(t +T ) = fi(t), i = 1, . . . p t > 0

(4)

let the five-tuple (Ai,Bi,Ci,Ei,Fi) represent the ith

plant model Pi of the family Π and the five-tuple(Aa,Ba,Ca,Ea,Fa) represent the actual system Pa, theregulation of which is the control objective. Sampling theerror signal will result in,

u[k] = e[k] = yre f [k]− y[k] (5)

The control signal will then be

u(t) = fi(t).[yre f [k]− y[k]

]

= fi(t).[yre f [k]− [Cix[k]+ Fiω[k]]

]

= fi(t).yre f [k]− fi(t).Cix[k]− fi(t).Fiω[k]

(6)

The corresponding sampled-data system for (3) is given bythe following equation,

x[(k +1)T ] = eAiT x[kT ]+∫ (k+1)T

kTeAi((k+1)T−t)Biu(t)dt

+∫ (k+1)T

kTeAi((k+1)T−t)Eiω(t)dt

(7)

Define

Ai := eAiT

Bi :=∫ T

0eAi(T−t)Bifi(t)dt

Ei :=∫ T

0eAi(T−t)Eidt

Ci := Ci

Fi := Fi

Assuming that the closed-loop system is stable and correctGSHF is being used, substituting u(t) from (6) into (7) willresult in,

x[k +1] = eAiT x[k]+∫ (k+1)T

kTeAi((k+1)T−t)Bi

.[fi(t).yre f [k]− fi(t).Cix[k]− fi(t).Fiω[k]]dt

+∫ (k+1)T

kTeAi((k+1)T−t)Eiω(t)dt

Hence, the discrete-time equivalent model for the resultantclosed-loop system will be given by,

x[k +1] = Aix[k]+ Biyre f [k]− BiCix[k]− BiFiω[k]+ Eiω[k]

= (Ai − BiCi)x[k]+ Bi(yre f [k]− Fiω[k])+ Eiω[k](8a)

y[k] = Cix[k]+ Fiω[k] (8b)

Having x[k +1] = φx[k]−ψ[k] we can write,

x[k] = φ kx[0]−k−1

∑i=0

φ k−i−1ψ[i] (9)

So the state of the closed-loop system at each samplinginstant can be related to the initial state and input signalsthrough the following equation,

x[k] = (Ai − BiCi)kx[0]

+k−1

∑ν=0

[(Ai − BiCi)k−ν−1[Bi(yre f [ν ]− Fiω[ν ])+ Eiω[ν ]]

]

(10)

Since we have assumed that the unmeasurable disturbancesignal is bounded, for the bound on the disturbance let’sassume that we have

ω := maxi

‖ω(i)‖

III. MAIN RESULTS

We need to prove the following inequality which will beused later in the proof of Lemma1.

Preliminary: Assume that y[k] = y1[k]+y2[k] hence we canwrite

‖y2[k]‖+‖y[k]‖ � ‖y2[k]− y[k]‖ = ‖y1[k]‖‖y1[k]‖2 � ‖y[k]‖2 +‖y2[k]‖2 +2‖y[k]‖.‖y2[k]‖

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we also know that (‖y2[k]‖−‖y[k]‖)2 � 0, By adding thislater inequality to the former we can obtain,

‖y1[k]‖2 � 2‖y[k]‖2 +2‖y2[k]‖2 (11)

This will later be used in the proofs.To make sure that the switching mechanism acts properly,

we will have to find a bound on the initial condition of thesystem. Same as the algorithm in [3] our method will beperformed in two phases: before applying the control signaland after that. The following lemma gives a bound on thenorm of the initial state of the system with the assumptionthat the disturbance is bounded and the control input is setto zero.

Lemma 1: Consider the system (8) and assume that thecontrol input u(t) in (3) is equal to zero for 0 � t � mT , forany arbitrary integer m > 0. There exists constants α1,i andα2,i, such that for any initial condition x[0] and disturbanceω , the following inequality on the norm of the initial stateholds:

‖x[0]‖2 ≤ α1,i

m

∑k=0

‖y[k]‖2 +α2,iω2 (12)

Proof of Lemma 1: The samples of the output obtained fromthe discrete equivalent system the output is given by:

y[k] = Cix[k]+ Fiω[k] (13)

Since the control input is set to zero for 0 � t � mT , it canbe concluded from (10) that

x[k] = Aki x[0]+

k−1

∑ν=0

[Ak−ν−1i Eiω[ν ]] (14)

Substituting (14)in (13) will result in:

y[k] = Ci[Aki x[0]+

k−1

∑ν=0

Ak−ν−1i Eiω[ν ]]+ Fiω[k]

= CiAki x[0]+Ci

k−1

∑ν=0

Ak−ν−1i Eiω[ν ]+ Fiω[k]

(15)

Define

y1[k] := CiAki x[0] (16a)

y2[k] := Ci

k−1

∑i=0

Ak−i−1i Eiω[i]+ Fiω[k] (16b)

Taking norm of the terms in both sides of (16b) results in:

‖y2[k]‖ � ‖ω[k]‖.(k−1

∑ν=0

‖CiAk−ν−1i Ei‖+‖Fi‖)

� ω.(k−1

∑ν=0

‖CiAk−ν−1i Ei‖+‖Fi‖)

(17)

Define now:

Wi :=m

∑k=0

(A′i)

kC′iCiA

ki

thenm

∑k=0

‖y1[k]‖2 = x[0]′Wix[0]

Now let

α3,i := smallest singular value of Wi (18)

It can be concluded from (16a) that:m

∑k=0

‖y1[k]‖2 −α3,i‖x[0]‖2 = x[0]′[Wi −α3,iI]x[0] � 0

m

∑k=0

‖y1[k]‖2 � α3,i‖x[0]‖2(19)

On the other hand, it follows from the preliminary results in(11), that

m

∑k=0

‖y1[k]‖2 � 2m

∑k=0

‖y[k]‖2 +2m

∑k=0

‖y2[k]‖2 (20)

Substituting y1[k] from (19) in (20) and dividing both sidesby α3,i will result in:

‖x[0]‖2 � 2α3,i

m

∑k=0

‖y[k]‖2 +2

α3,i

m

∑k=0

‖y2[k]‖2 (21)

Now define α1,i and α2,i as follows

α1,i :=2

α3,i

α2,i :=2

α3,i

m

∑k=0

[k−1

∑ν=0

‖CiAk−ν−1i Ei‖+‖Fi‖]2

Then using (17), one can obtain the following inequality,

‖x[0]‖2 ≤ α1,i

m

∑0‖y[k]‖2 +α2,iω2

�Lemma 1 states that if a bound ω on the norm of

disturbance ω exists, then one can find a bound on the normof the initial condition as long as the control input is zero.

It is desired now to obtain an upper bound on the state ofthe system at each sampling instant, and at the presence ofthe control signal. This bound will be used in the decisionmaker unit to verify weather the current GSHF is correct orit needs to be changed. Since the closed-loop system is notnecessarily stable hence the equation obtained in (10) cannot be used for x[k], therefore we need to find x[k] relatedto the initial state and the input signal, even if the correctGSHF is not being applied to the system, Having (7) and (4)we can write,

x[k +1] = eAiT x[k]

+∫ (k+1)T

kTeAi((k+1)T−t)Bi.[f j(t).u[k]]dt

+∫ (k+1)T

kTeAi((k+1)T−t)Eiω(t)dt

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To find the bound on the state even if the GSHF used in thesystem is not the correct one (This happens while the systemswitches to different GSHFs to find the correct one, DefineBi, j as the discrete-time equivalent matrix for Bi, when theGSHF f j corresponding to the plant P j is used in the closed-loop system. Bi, j can be obtained as follows:

Bi, j :=∫ T

0eAi(T−t)Bif j(t)dt

Apparently, when i = j we have Bi, j = Bi hence,

x[k +1] = Aix[k]+ Bi, j u[k]+ Eiω[k]

= (Ai − BiCi)x[k]+ BiCix[k]+ Bi, j u[k]+ Eiω[k]

= (Ai − BiCi)x[k]+ Bi[y[k]− Fiω[k]]+ Bi, j u[k]+ Eiω[k]

according to (9) we can now write:

x[k] = (Ai − BiCi)kx[0]

+k−1

∑ν=0

[(Ai − BiCi)k−ν−1[Biy[ν ]+ Bi, j u[ν ]+ (Ei − BiFi)ω[ν ]]

]

(22)

The following lemma gives an upper bound on the stateof the system in terms of the system parameters and normsof the disturbance ω and reference signal yre f denoted by ωand yre f respectively.

Lemma 2: There exists constants γ1,i ,γ2,i and λi such thatfor the control signal u(t) and bounded disturbance ω andinitial state x[0] the following inequality holds:

‖x[k]‖ � λiγ1,ik‖x[0]‖

+k−1

∑ν=0

[λiγ1,i

k−ν−1[‖Biy[ν ]‖+‖Bi, j u[ν ]‖+ γ2,i]] (23)

Proof of Lemma 2: It follows from (22) that:

‖x[k]‖ � ‖(Ai − BiCi)k‖‖x[0]‖+k−1

∑ν=0

[‖(Ai − BiCi)k−ν−1‖

.[‖Biy[ν ]‖+‖Bi, j u[ν ]‖+ ω‖Ei − BiFi‖]]

(24)

A proper λi can be found such that

‖(Ai − BiCi)k‖ � λi.max |eig(Ai − BiCi)|k

by choosing γ1,i = max |eig(Ai − BiCi)| and γ2,i = ω‖Ei −BiFi‖ we can prove the lemma, and hence find a norm onthe state of the system.

�Having γ3,i = ω‖Ci(Ei − BiFi)‖ a bound on the norm of

the output can now be found by using (13) as follows:

‖y[k]‖ � ‖Ci‖λiγ1,ik‖x[0]‖+

k−1

∑ν=0

[λiγ1,i

k−ν−1

.[‖CiBiy[ν ]+CiBi, j u[ν ]‖+ γ3,i]]]

+ ω‖Fi‖(25)

Assuming that the disturbance is bounded, By substitutingthe bound on the initial condition given by Lemma 1 into(25), one can find an auxiliary signal which gives a boundon the state of the system at any point of time. let ri denotethe auxiliary signal which is a bound on the norm of theoutput of the closed-loop system corresponding to the plantmodel Pi. In other words ri is a bound on the output of theplant Pi, regardless of which GSHF is used to control thesystem. This auxiliary signal is given by:

ri[k] = ‖Ci‖λiγ1,ik‖x[0]‖

+k−1

∑ν=0

[λiγ1,i

k−ν−1.[‖Ci(Biy[ν ]+ Bi, j u[ν ])‖+ γ3,i]]+ ω‖Fi‖

ri[k +1] = ‖Ci‖λiγ1,ik+1‖x[0]‖

+k

∑ν=0

[λiγ1,i

k−ν .[‖Ci(Biy[ν ]+ Bi, j u[ν ])‖+ γ3,i]]+ ω‖Fi‖

(26)

In order to come up with a difference equation to update theauxiliary signals at each step one can write,

ri[k +1] = γ1,i.ri[k]+λi[‖Ci(Biy[k]+ Bi, j u[k])‖+ γ3,i]

+ (1− γ1,i)ω‖Fi‖(27)

Note that,

ri[0] = ‖Ci‖.λi‖x[0]‖+ ω‖Fi‖Using the results of the Lemma 1 and substituting the upperbound on the initial state into the above equation, for 0 �t � mT , and for any arbitrary integer m > 0 with no input,results in:

ri[t] � λi‖Ci‖.[α1,i

m

∑k=0

‖y[k]‖2 +α2,iω2]12 + ω‖Fi‖

Choosing proper constants to satisfy the inequalities inLemma 1 and Lemma 2, one can break the control processinto two phases, as follows,

Phase 1: Setting the control signal to zero for 0 � t � mTwhere m is any non-zero integer and T is the sampling periodresults in the following upper bound on the norm of theoutput of the system.

ri[k] = λi‖Ci‖.[α1,i

m

∑ν=0

‖y[ν ]‖2 +α2,iω2]12 + ω‖Fi‖

Phase 2: The equation (27) can now be used to updatethe new values of the auxiliary signals for k > m. It is tobe noted that the norm of the output is compared with thecorresponding upper bound only at the sampling instants (thediscrete model will be used in the decision making unit ofthe switching control.)

It is now desired to find the switching instants using theauxiliary signals obtained above.

Switching Controller 1: Set t1 = mT , and for every i ∈{2, . . . , p+1} for which ti−1 �= ∞, define:

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ti := min{t � ti−1,∃k ∈ [mT, t] | ‖y[k]‖ � ri[k]}Theorem 1: Suppose that the reference input and the

disturbance signals are piecewise constant and bounded, i.e.‖ω(t)‖ � ω and yre f (t) � yre f for t � 0. For every initialcondition x(0), when the Switching Controller 1 is appliedto the uncertain plant, where model belongs to the knownfamily of plant models Π the closed loop system has thefollowing properties:a) the GSHF will ultimately remain unchanged at an elementof {f j; j ∈ p} and;b) the state of the system will be bounded.

Proof : It was shown in (25) that ‖y[k]‖� ri[k] for k ≥ mT ,Thus, it can be concluded from Switching Controller 1, thatti+1 must be ∞. Hence part (a) holds. On the other hand,boundedness of the state of the system follows immediatelyfrom Lemma 2. �

IV. SIMULATION RESULTS

Consider the two cart mass-spring-damper system of Fig-ure 2 as shown in [13]. The control force is applied to themass m1, and the output of the system is the position of thesecond cart (m2). The state-space model is represented bythe following matrices:

m1 m2

Control

u(t)

x1 x2

k

c

Fig. 2. The two-cart SISO system connected with spring and damper

A =

⎡⎢⎢⎣

0 0 1 00 0 0 1−km1

km1

−cm1

cm1

km2

−km2

cm2

−cm2

⎤⎥⎥⎦

BT =[0 0 1

m10]

C =[0 1 0 0

]

where k and c are the spring constant and the dampingcoefficient, respectively. It is assumed that the disturbancesignal ω(t) is equal to zero, and that the reference input isthe unit step signal. Consider a family of four plant modelsgiven by:

P1 : m1 = 6.2032, m2 = 5.0660,k = 1.0011, c = 0.0104

P2 : m1 = 7.8113, m2 = 9.5371,k = 1.1226, c = 0.2168

P3 : m1 = 5.9745, m2 = 8.9869,k = 0.8837, c = 0.1359

P4 : m1 = 2.1017, m2 = 1.2885,k = 0.5548, c = 0.0561

A GSHF is designed for each plant model using the methodproposed in [9].

Assume now that the actual plant model was initiallyP4 and at some point of time it suddenly changes to P2.Using the proposed switching mechanism given by SwitchingController 1 the system will first switch from f4 to f1 and thento f2. The system locks onto f2 as it is the only GSHF thatstabilizes P2, which is the new plant model. The switchinginstants are shown in Figure 3. The output of the system isgiven in Figure 4. This figure shows good regulation for thegiven system parameters.

0 1 2 3 4 5 6 7 8 9 10

f1

f2

f3

f4

Fig. 3. Switching instants, when the plant model changes from P4 to P2

0 10 20 30 40 50 60 70 80 90 100

−120

−100

−80

−60

−40

−20

0

20

40

60

Fig. 4. Output signal of the closed-loop simulation results, when the plantmodel changes from P4 to P2

V. CONCLUSIONS

In this paper, a switching control mechanism using gen-eralized sampled-data hold functions (GSHF) is proposedto regulate an uncertain plant whose model belongs to a

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finite set of known plant models. It is assumed that aset of GSHFs are designed off line such that each GSHFstabilizes only one of the plant models in the given set. Theswitching mechanism is based on the discrete-time equivalentmodel and a discrete-time upper bound signal which iscompared to the samples of the output at each samplinginstant. It is shown that the system eventually switches to thecorrect GSHF and locks onto it, provided that bounds on thereference signal and unmeasurable disturbance are given. Theproposed method is computationally more efficient comparedto the continuous-time counterparts. Simulation results showeffectiveness of the method for a practical mass-spring-damper system.

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[6] A. B. Chammas and C. T. Leondes, “On the finite time control oflinear systems by piecewise constant output feedback,” InternationalJournal of Control, vol. 30, no. 2, pp. 227–234, 1979.

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[10] A. G. Aghdam and E. J. Davison, “Decentralized control of systems,using generalized sampled-data hold functions,” Proceedings of the38th IEEE Conference on Decision and Control, vol. 4, pp. 3912–3913, Dec. 1999.

[11] A. G. Aghdam and E. J. Davison, “Application of generalized sampled-data hold functions to decentralized control structure modification,”Proceedings of the IFAC 15’th World Congress, Barcelona, Spain, AreaCode 5a, Session Slot T-Th-A13, July 2002.

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