ON THE DESIGN OF A QUANTIZED STATE-FEEDBACK CONTROL ... · Email:...

ON THE DESIGN OF A QUANTIZED STATE-FEEDBACK CONTROL

Vlademir Aparecido Freire Junior, Alessandro do Nascimento Vargas, CristianoMarcos Agulhari

∗Laboratorio de Controle e Otimizacao de Sistemas, Universidade Tecnologica Federal do ParanaAvenida Alberto Carazzai, 1640 - Centro - Cornelio Procopio, Parana, Brasil

Email: [email protected],[email protected],[email protected]

Abstract— This paper presents a new technique to synthesize stabilizing state-feedback control laws for lineartime-invariant systems, considering that the feedback is performed using quantized states. To tackle such problem,the quantization error is initially modeled as an external noise. By doing so, the calculation of state-feedbackgains that are robust to the quantization effects becomes equivalent to obtaining the gains that minimize the H∞norm of the closed-loop system. The stabilizing gains are calculated through the resolution of a linear matrixinequality condition proposed by the authors. The efficacy of the presented technique is illustrated by applicationof the synthesized state-feedback on a servo-mechanism didactic module.

Keywords— Quantized Control, H∞ Norm, LMIs, Robust Control.

Resumo— Este artigo apresenta uma nova tecnica para a sıntese de leis de controle estabilizantes de reali-mentacao de estados para sistemas lineares invariantes no tempo, considerando que a realimentacao e realizadautilizando estados quantizados. Para abordar tal problema, o erro de quantizacao e modelado inicialmente comoum ruıdo externo. Ao fazer isso, o calculo dos ganhos de realimentacao de estados que sao robustos aos efeitosde quantizacao torna-se equivalente a obtencao dos ganhos que minimizam a norma H∞ do sistema em malhafechada. Os ganhos de estabilizacao sao calculados atraves da resolucao de uma condicao de desigualdade ma-tricial linear proposta pelos autores. A eficacia da tecnica apresentada e ilustrada pela aplicacao do estado darealimentacao sintetizados em um modulo didatico servo-mecanismo.

Keywords— Controle Quantizado, Norma H∞, LMIs, Controle Robusto.

1 Introduction

Considerable efforts have been devoted to thestudy of quantized control systems, due to its the-oretical and practical importance in the study ofdigital control (Che and Yang, 2007) and due toits flexibility, remote control, reduced wiring andlow maintenance cost (Rasool and Nguang, 2010).One of the main challenges on digital control con-cerns the conversion of the analogic signals, whichare the inputs and outputs of the system, to a dig-ital representation. There are mainly two steps insuch conversion: sampling and quantization. Re-garding the latter, there is a tradeoff between thenumber of quantization levels considered (directlyrelated to the number of bits used to representeach sample) and the quality of representation.There are several studies that seek the minimumnumber of bits (or the minimum number of quan-tization levels) required to guarantee the stabilityof the system (Wong and Brockett, 1999; Ling andLemmon, 2005; Hespanha et al., 2007; Elia andMitter, 2001), despite of the uncertainties intro-duced such as time delays and quantization errors(Zhang et al., 2001; Zhang et al., 2008).

In (Liu et al., 2013), it is presented a tech-nique, based on LMIs (Linear Matrix Inequali-ties), to synthesize state-feedback gains consider-ing that the states are subjected to an uniformquantizer. Such technique, however, does not syn-thesize a state-feedback gain that minimizes theeffect of quantization of the states in the closed-loop system; instead, a state-feedback gain is com-

puted and the robustness of the controller to thequantization is verified a posteriori. In (Brockettand Liberzon, 2000), the quantization error ismodeled as an external noise, and the controlleris synthesized to guarantee the robustness of thesystem to the noise. The technique, however, re-quires the possibility of dinamically changing thenumber of quantization levels, which increases sig-nificatively the complexity of the procedure in apractical situation. The majority of techniques inthe literature (e.g. (Che and Yang, 2007; Rasooland Nguang, 2010; Fu and Xie, 2005)) considerlogarithmic quantizers and deal with discrete-timesystems, since both considerations promote someinteresting properties that may ease the synthesisprocedures.

The method proposed in this paper is capa-ble of synthesizing quantized state-feedback gainsfor continuous-time systems, robust to the quan-tization of the states up to a certain numberof quantization levels, considering the utilizationof uniform quantizers. The technique is similarto the strategy adopted in (Brockett and Liber-zon, 2000). First, the effect of constraining thesystem states to a quantizer is modeled as the in-sertion of an external noise on the system. Usi-ng such model, it is proposed an LMI-based H∞(Boyd et al., 1994; Bernussou et al., 1989) tech-nique to generate controllers that minimizes theeffects of quantization on the states. The immedi-ate advantage of this technique is that it is some-what easy to be applied, and the H∞ norm boundcan be directly related to the number of bits of

Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014

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quantization that can be used without unstabi-lizing the system. An experiment on a practicalsystem illustrates the advantages of the proposedcontrol technique.

The paper is organized as follows. In Sec-tion 2 the preliminary results are presented. Themain results of the work are presented in Section3. Section 4 presents the experimental results thatillustrate the validity of the technique and Section5 concludes the paper.

2 Preliminaries

Consider the continuous time-invariant linear sys-tem described by

x(t) = Ax(t) +Bu(t), (1)

where x(t) ∈ Rn represents the state vector,u(t) ∈ Rm represents the control input, and A ∈Rn×n, B ∈ Rn×p are fixed matrices that describethe dynamics of the system.

The main problem investigated here concernsthe stabilization of system (1) using a quantizedstate-feedback control law, i.e., the problem of cal-culating a fixed state-feedback gain K ∈ Rm×nsuch that the control law

u(t) = K∆q (x(t)) (2)

stabilizes asymptotically the system, being ∆the quantization step of the uniform quantizerq (x(t)), defined as

q (x(t)) =

0, if

⌊x(t)∆

⌋< 1

∆(⌊

x(t)∆

⌋+ 1

2

), otherwise,

(3)

where b.c denotes the largest integer, smaller thanx(t)/∆.

More details on uniform quantizers may befound in (Sayood, 2012). The application of u(t)as in (2) on system (1) yields

x(t) = Ax(t) +BK∆q (x(t)) .

Considering the quantization error w(x(t))given by

w(x(t)) = x(t)−∆q(x(t)),

one has

x(t) = (A+BK)︸︷︷︸Acl

x(t)− BK︸︷︷︸Bcl

w (x(t)) . (4)

Note that |w(x(t))| ≤ ∆/2 for all x(t). Asa consequence, although the proposed control lawu(t) is nonlinearly dependent on the states, the

closed-loop system (4) can be seen as an LTI sys-tem affected by a bounded external disturbancew(x(t)). Consider the following system

G(t)

{x(t) = Aclx(t)−Bclw(t)z(t) = Cx(t),

(5)

being z(t) ∈ Rq the controlled output and C ∈Rq×n the observation matrix. From (Boyd et al.,1994), the stability of (5) is more robust to theexternal noise w(x(t)) if the H∞ norm of the sys-tem is minimized. Since |w(x(t))| ≤ ∆/2, thismeans that lower values for the H∞ norm allowshigher values for the quantization step ∆ with-out compromising the closed-loop stability, on theutilization of less bits in the quantization step.

In the following lemma, it is presented an LMIcondition for the calculation of a bound to theH∞norm of a given system.

Lemma 2.1 (Bounded Real Lemma)System (5) is asymptotically stable and itsH∞ norm is lower than a given positive scalar γif, and only if, there exists a symmetric positivedefinite matrix P ∈ Rnxn such that[

A′clP + PAcl + C ′C −PBcl−B′clP −γ2I

]< 0. (6)

The minimum value for γ can be calculated bysolving the following optimization problem

γ∗ = min γ s.t. (6) (7)

In the usual H∞ state-feedback synthesisproblem, only the Acl matrix is dependent on thestate-feedback gain K, and simple manipulationscan be used to derive convex conditions to cal-culate the desired control gain. However, in thequantized state-feedback case considered in thispaper, both matrices Acl and Bcl depends on K,which means that a different approach should beconsidered.

Let D be a sub-region of the complex planeC. The following definitions and theorems can beused to verify if the poles of a LTI system areinside the region D.

Definition 2.1 Let R ∈ R2d×2d be a partitionedsymmetric matrix as

R =

[R11 R12

R′12 R22

]:R11 = R′11 ∈ Rd×d,R22 = R′22 ∈ Rd×d.

being d denotes the order of the region (Leite andPeres, 2003). A region D in the complex planecan be described as

D = {z ∈ C : R11 +R12z +R′12z∗ +R22zz

∗ < 0}.(8)


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Figure 1: Graphical representation of D-Stability.

Assuming that R22 ≥ 0, one has that D isa convex region and symmetric in relation to thereal axis (Bernussou et al., 1989).

Definition 2.2 The matrix A ∈ Rn×n is D-stableif, and only if, all the eigenvalues are in the regionD defined in (8).

Proposition 2.1 (Peaucelle et al., 2000) A ∈Rn×n is D-stable if, and only if, there is a symme-tric positive definite matrix P ∈ Rn×n such that

R11⊗P+R12⊗(PA)+R′12⊗(A′P )+R22⊗(A′PA) < 0.

(9)

with ⊗ being the Kronecker product.

Generically, the stability region is the left halfplane for continuous time systems. This region issimply given by

Rcont =

[0 11 0

].

Figure 1 illustrates the region D when consi-dering R = Rcont.σ, being σ a positive scalar.

Using such matrix R on (9) one has the con-dition

Re {λi (A+ σ)} < 0, i = 1, ..., nm

∃P = P ′ > 0 : (A+ σI)′P + P (A+ σI) < 0.

(10)Therefore, if condition (10) is satisfied, then

the real part of all the eigenvalues of A are lowerthan σ.

The following theorem presents the result thatwill be used to establish the relation between theH∞ norm of the system and the minimum numberof bits of quantization that guarantees the stabi-lity of the controlled system.

Theorem 2.1 (Small-Gain Theorem)Consider the system G and the external noiseδ illustrated by Figure 2, and suppose that G isinternally stable. If the norms of G and δ aresuch that

‖δ‖∞‖G‖∞ < 1, (11)

then system G remains stable despite of the exter-nal noise δ.

Therefore, the small gain theorem plays animportant role in robust control theory, for sys-tems affected by external and bounded distur-bances (Green and Limebeer, 2012).

3 Main Results

The following theorem presents a sufficient condi-tion to obtain a state-feedback stabilizing gain forsystem (5), in such a way that the H∞ norm ofthe closed-loop system is bounded by a predefinedscalar γ.

Theorem 3.1 If there exist a symmetric positivedefinite matrix W ∈ Rnxn, a matrix Z ∈ Rmxn

and a positive scalar γ such that the condition

WA′ +AW + Z′B′ +BZ −BZ WC′

∗ I− 2γW 0∗ ∗ −I

< 0

(12)

is verified, then the control law u(t) = K∆q (x(t)),with K = ZW−1, stabilizes system (5) with aguaranteed H∞ norm given by γ.

Proof: Applying the closed-loop matrices of sys-tem (5) on condition (6), and after the applicationof Schur’s complement (Boyd et al., 1994), one has

A′P + PA+K′B′P + PBK −PBK C′

∗ −γ2I 0∗ ∗ −I

< 0.

(13)

Multiplying condition (13) on the left by Tand on the right by its transpose, where

T =

P−1 0 00 P−1 00 0 I

,one gets


∗ −γ2W 2 0∗ ∗ −I

< 0,

(14)

Figure 2: The small gain theorem.


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with W = P−1. In order to linearize inequa-lity (14), note that

(I− γW )′(I− γW ) ≥ 0,

which implies that

−γ2W 2 ≤ I− 2γW,

therefore, the matrix in (14) is bounded above by


∗ I− 2γW 0∗ ∗ −I

.(15)

Consequently, if (15) is definite negative, thencondition (6) is satisfied. According to Lemma2.1, this means that the closed-loop system withu = K∆q (x) is asymptotically stable with H∞norm lower than γ. 2

Remark 3.1 Note that condition (12) is not anLMI, due to the nonlinear block 2 × 2. However,if γ is a predefined scalar value, then the condi-tion (12) becomes linear.

Based on the value of γ considered in Theo-rem 3.1, following corollary presents a conditionto calculate the lower value of bits of quantizationthat still guarantees the stability of system (5).

Corollary 3.1 If the number N of bits of thequantizer in (3) is such that

N > log2

(Vmaxγ

2

)(16)

being Vmax the maximum output value of thequantizer, then the closed-loop system (7) with thestate-feedback gain as in Theorem 3.1 is stable.

Proof: Replacing w and γ in (11), one has

‖w‖‖γ‖ < 1.

Since ‖w‖ < ∆/2, then

‖w‖‖γ‖ <∆

2γ < 1 ⇒ ∆ <

2

γ. (17)

The value of ∆ is calculated by

∆ =Vmax

2N, (18)

being N the number of bits of the converter. Re-placing (18) in (17), one arrives in the followingcondition

N > log2

(Vmaxγ

2

). (19)

Thus (19) shows the lower value of bits thatguarantees the stability of the system. 2

Figure 3: Servo-mechanism module didactic.

If it is desired that the closed-loop system,resultant from the application of the gain obtainedthrough Theorem 3.1, has eigenvalues with realpart lower predefined σ one can use the followingtheorem.

Theorem 3.2 If there is a matrix W = W ′ ∈Rnxn positive definite, a matrix Z ∈ Rmxn and ascalar γ > 0 such that the LMI

W (A+ σI)′ + (A+ σI)W + Z ′B′ +BZ∗∗

−BZI− 2γW∗

WC ′

0−I

< 0. (20)

is verified, then control law u(t) = K∆q (x(t)),with K = ZW−1, stabilizes the system (5) with anorm H∞ limited by γ and the eigenvalues of thematrix A of closed loop less than σ.

Proof: The condition is obtained by applying thesame mathematical manipulation, performed onTheorem 3.1, over the condition (10). 2

4 Experimental Results

In order to illustrate the usefulnes of the proposedtechnique, a practical experiment on a plant wasperformed. The LMIs were solved, using Yalmip(Lofberg, 2004) and SeDuMi (Sturm, 1999) underMatlab (R2010a). The plant is the Module DCMotor 2208, made by Datapool Eletronica Ltda,Brazil, and it is shown in Figure 3.

To measure the angular velocity, we usedthe manufacturer-provided tachogenerator, whichproduces a voltage proportional to the velocity ofthe shaft. To measure the electric current, we de-veloped a simple conditioning circuit which is in-troduced in series with the motor, composed by a


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shunt resistor connected with a pre-amplifier sig-nal stage. The respective matrices A and B thatdescribe the open-loop behavior of the system (1)were identified and the corresponding values aregiven below:

A =

[−0.9315 6.8974−1.4293 −6.6652

], B =

[0

10.1463

](21)

The objective of the experiment is to controlthe devide so as to follow a velocity reference inputsignal. The considered controller includes an inte-gral action, in order to cancel the steady-state er-ror (Ostertag, 2011), and is described in the blockdiagram shown in Figure 4.

Figure 4: Block diagram of the system integrator.

In Figure 4, r(t) is the reference, e(t) is theerror, η(t) is the integral of the error, K =[K1 K2 K3] are the feedback gains, ω(t) (angularvelocity (rad/s)) and i(t) (motor current (A)) arethe state variables and y1(t) is the angular velocityoutput, which is calculated by

y1(t) = Cx(t), C =[

1 0].

The state-space representation of the aug-mented system, considering η(t) as an additionalstate, is given by

ddt

ω(t)i(t)η(t)

=

[A+B[ K1 K2 ] B [K3]

−C 0 0

]︸︷︷︸

Acl

ω (t)i (t)η (t)

−[−B

[K1 K2

]0

]︸︷︷︸

Bcl

w (t) +

001

r (t) .

(22)

The acquisition of the control signals was car-ried out by a data acquisition board from NationalInstruments NI USB-6008 model, with analog in-puts/outputs of 12 bit of resolution. The quan-tization effect is artificially introduced using (3),and the number N of bits considered were 12, 10,8 and 6 bits. Since the maximum voltage appliedto the module is Vmax = 5Volts, the values ∆ are

∆ =Vmax

2N=

0.0012 if N = 120.0049 if N = 100.0195 if N = 80.0781 if N = 6

where N is the number of bits of the quantizer.

−4 −3 −2 −1 0 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Figure 5: Allocation of the eigenvalues of the ini-tial system.

First, the state-feedback gains are calculatedthrough the minimization of γ subject to (12).The optimization has been performed using the bi-section technique (Bazaraa et al., 2013), since (12)is not linear (Remark 3.1).

The optimal gains corresponding to the state-feedback obtained through the optimization (12)are

K = [−0.0055 0.0053 0.0001] , (23)

with a bound γ∗ = 0.22. Though, the use of thevalues of the gains and γ∗ makes the system robustto quantization, however, results in a very slowresponse as follows. Indeed, replacing the matricesA e B presented in (21) with (23) in (22), one has

Acl =

−0.9315 6.8974 0−1.4851 −6.6114 0.0010−1 0 0

and

Bcl =

0 00.0558 −0.0538

0 0

,whose eigenvalues are illustred on Figure 5.

The system presented a slow response due tothe presence of one eigenvalue close to the ima-ginary axis (Ogata, 2000). This problem can bemitigated using the Theorem 3.2.

The value of σ, which will delimit the regionwhere the eigenvalues of the matrix Acl, repre-sented by λmax, can be allocated, is defined con-sidering a criterium to the settling time of the an-gular velocity. In particular, it is defined a pri-ori that the angular velocity must reach at least90% of the input reference value at a given settlingtime. Therefore, one has to calculate λmax suchthat

1− eλmaxta ≥ Vref , (24)


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−4 −3 −2 −1 0 1−4

−3

−2

−1

0

1

2

3

4

σ

Figure 6: Allocation on of the eigenvalues of thesystem using the D-Stability.

where ta represents the settling time and Vrefthe value of the reference input. Considering asproject criterions Vref = 1 rad/s and ta = 4 se-conds, replacing in (24) and isolating λmax, onehas

λmax ≤ −0.5756.

Solving Theorem 3.2 with the restrictions ofD-Stability with σ = 0.5756, the resulting statefeedback gains are

K = [−0.1838 − 0.0694 0.3485] , (25)

with the value of γ∗ = 1.8. Replacing the matricesA and B given in (21) with (25) in (22) results

Acl =

−0.9315 6.8974 0−3.2938 −7.3691 3.5362−1 0 0

and

Bcl =

0 01.8645 0.7039

0 0

.Figure 6 shows the arrangement of the eigen-

values of Acl displaced with respect to the ima-ginary axis of the complex C plane, showing thecorrectness of the result.

To analyze the response performance of theproposed system, it is applied a reference signalon the servo-mechanism module. Through of therespective Figures 7, 8, 9, 10, we can see that theproposed controller resulted in a response signalthat has no maximum overshoot and showing lit-tle variation in the control law. The performanceof the control laws using 12-bit quantization areshown in Figure 7.

0 20 40 60 80 100 120 1400

1

2

3

4

Angular Velocity

Vel

ocity

(ra

d/s)

Velocity

Reference

0 20 40 60 80 100 120 1400

0.5

1

Control Effort

Time (s)

Vol

ts(V

)

Figure 7: Response curve control, quantized in12-bit.

It is possible to verify that the proposed con-troller showed a considerable robustness to thequantization, resulting on a signal without a ma-ximum overshoot and with little variation in thecontrol law. In Figure 8, the results considering10-bit quantization are presented. In the thirdanalisys, the control signals are quantized with 8-bit, as shown in Figure 9. Again, it is possible toverify the robustness of the H∞ controller.

One can verify that through the proposed con-trol showed a greater variation on speed the res-ponse, reached the reference values. Note that,as the number of bits decreased, worse responseswere obtained. However, the proposed controlshowed robust responses to the effects of quan-tization error on all analysis, reaching the refe-rence input and requiring low control effort. Fi-gure 10 shows the results obtained when using a6-bit quantization.

According to Corollary 3.1, the lower num-ber of bits such that the quantized state-feedbackgain (25) still ensures the stability of the closed-

0 20 40 60 80 100 120 1400

1

2

3

4

Angular Velocity

Vel

ocity

(ra

d/s)

Velocity

Reference

0 20 40 60 80 100 120 1400

0.5

1

Control Effort

Time (s)

Vol

ts(V

)



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0 20 40 60 80 100 120 140 1600

1

2

3

4

Angular Velocity

Vel

ocity

(ra

d/s)

Velocity

Reference

0 20 40 60 80 100 120 140 1600

0.5

1

Control Effort

Time (s)

Vol

ts(V

)

Figure 9: Response curve control, quantized in 8-bit.

0 20 40 60 80 100 1200

1

2

3

4

Angular Velocity

Vel

ocity

(ra

d/s)

Velocity

Reference

0 20 40 60 80 100 1200

0.5

1

Control Effort

Time (s)

Vol

ts(V

)


loop system is N = 3 bits. Experiments using3 bits of quantization indicated that the outputsignal followed the reference as expected, but itwas highly oscillatory and, therefore, improper forpractical applications as Figures 11 and 12.

On the other hand, if N < 3 bits are used, theresponse is not a divergent one. In fact, the state-feedback gain guarantees the stability for N ≥ 3bits, but nothing can be concluded for N < 3.

The technique proposed in this paper has beencompared to the method of (Liu et al., 2013),whose result for N = 12 bits is shown in Fi-gure 13. Note that, even for a high number of bits,the performance is far inferior when compared toFigure 7. More details on the implementationof (Liu et al., 2013) can be found in (Freire, 2014).

5 Conclusions

A technique to synthesize state-feedback gains, ro-bust to the effect of uniform quantization on thestates, is proposed. Since the closed-loop systemwith quantized state-feedback can be modeled as

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time (s)

Vel

ocity

(ra

d/s)

VelocityReference


0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Vel

ocity

(ra

d/s)

Time (s)

Velocity

Reference


0 20 40 60 80 100 120 1400

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time (s)

Vel

ocity

(ra

d/s)

VelocityReference

Figure 13: Response curve control propose by (Liuet al., 2013) , quantized in 12-bit.

a linear time-invariant system affected by an ex-ternal noise, the problem can be seen as the syn-


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thesis of a state-feedback gain that minimizes theH∞ norm of the closed-loop system, see Theo-rem 3.1. It is also shown the relation betweenthe H∞ norm and the lower acceptable numberof bits of quantization applied on the states ofthe system. The control law is calculated by theresolution of an LMI condition, Theorem 3.2. Ex-perimental results illustrate the efficiency of thetechnique, stabilizing the system even with a lownumber of quantization bits.

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