Target mass control for uncertain compartmental systems

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<ul><li><p>This article was downloaded by: [North Dakota State University]On: 14 November 2014, At: 08:05Publisher: Taylor &amp; FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK</p><p>International Journal of ControlPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tcon20</p><p>Target mass control for uncertain compartmentalsystemsClaudia Sousa a , Teresa Mendonca b &amp; Paula Rocha ca School of Education Jean Piaget , Piaget Institute , R. Antonio Sergio, 4410-269, CanelasV.N.G., Portugalb Department of Applied Mathematics, Faculty of Sciences , University of Porto , R. doCampo Alegre, 687, 4169-007, Porto, Portugalc Department of Electrical and Computer Engineering, Faculty of Engineering , University ofPorto , R. Dr. Roberto Frias, s/n, 4200-465, Porto, PortugalPublished online: 19 May 2010.</p><p>To cite this article: Claudia Sousa , Teresa Mendonca &amp; Paula Rocha (2010) Target mass control for uncertain compartmentalsystems, International Journal of Control, 83:7, 1387-1396, DOI: 10.1080/00207171003736311</p><p>To link to this article: http://dx.doi.org/10.1080/00207171003736311</p><p>PLEASE SCROLL DOWN FOR ARTICLE</p><p>Taylor &amp; Francis makes every effort to ensure the accuracy of all the information (the Content) containedin the publications on our platform. However, Taylor &amp; Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor &amp; Francis. 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Terms &amp; Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions</p><p>http://www.tandfonline.com/loi/tcon20http://www.tandfonline.com/action/showCitFormats?doi=10.1080/00207171003736311http://dx.doi.org/10.1080/00207171003736311http://www.tandfonline.com/page/terms-and-conditionshttp://www.tandfonline.com/page/terms-and-conditions</p></li><li><p>International Journal of ControlVol. 83, No. 7, July 2010, 13871396</p><p>Target mass control for uncertain compartmental systems</p><p>Claudia Sousaa*, Teresa Mendoncab and Paula Rochac</p><p>aSchool of Education Jean Piaget, Piaget Institute, R. Antonio Sergio, 4410-269, Canelas V.N.G., Portugal;bDepartment of Applied Mathematics, Faculty of Sciences, University of Porto, R. do Campo Alegre, 687,4169-007, Porto, Portugal; cDepartment of Electrical and Computer Engineering, Faculty of Engineering,</p><p>University of Porto, R. Dr. Roberto Frias, s/n, 4200-465, Porto, Portugal</p><p>(Received 5 October 2009; final version received 27 February 2010)</p><p>In this article we analyse the total mass target control problem for compartmental systems under the presence ofparameter uncertainties. We consider a state feedback control law with positivity constraints tuned for a nominalsystem, and prove that this law leads the value of the total mass of the real system to an interval whose boundsdepend on the parameter uncertainties and can be made arbitrarily close to the desired value of the total masswhen the uncertainties are sufficiently small. Moreover, we prove that for a class of compartmental systems in R3</p><p>of interest, the state of the controlled system tends to an equilibrium point whose total mass lies within theaforementioned interval. Taking into account the relationship between the mass and the state components insteady state, it is possible to use the proposed mass control law to track the desired values for the steady statecomponents. This is applied to the control of the neuromuscular blockade level of patients undergoing surgery,by means of the infusion of atracurium. Our results are illustrated by several simulations and a clinical case.</p><p>Keywords: compartmental systems; positive control; uncertain systems; neuromuscular blockade control</p><p>1. Introduction</p><p>Compartmental systems form a subclass of positive</p><p>systems that consist of a finite number of subsystems,</p><p>the compartments, which exchange matter with each</p><p>other and with the environment. Such systems have</p><p>been successfully used to model biomedical and</p><p>pharmacokinetical processes; see, for instance,</p><p>Godfrey (1983) Jacquez and Simon (1993). Since one</p><p>has to guarantee the positivity of the control input, the</p><p>design of suitable control laws for such systems is more</p><p>delicate. In Haddad, Hayakawa, and Bayley (2003), for</p><p>instance, a nonnegative adaptive control law is pro-</p><p>posed in order to guarantee the partial asymptotic set-</p><p>point stability of the closed-loop system, and a positive</p><p>feedback control law is proposed in Bastin and Provost</p><p>(2002), in order to stabilise the total system mass at an</p><p>arbitrary set-point. The positive control law proposed</p><p>in Bastin and Provost (2002) was also used in</p><p>Magalhaes, Mendonca, and Rocha (2005) for the</p><p>control of the neuromuscular blockade level (Lemos,</p><p>Mendonca, and Mosca 1991; Linkens 1994; Mendonca</p><p>and Lago 1998) of patients undergoing surgery; how-</p><p>ever, no theoretical study was made of the effect of</p><p>parameter uncertainty in the controller performance.</p><p>In this article, we consider the control law referred</p><p>to in Bastin and Provost (2002) and in Magalhaes et al.</p><p>(2005), and analyse its performance for the target</p><p>control of the total mass, when the system parameters</p><p>are not exactly known. More concretely, we consider</p><p>that the control law is tuned for a nominal process</p><p>model that contains an additive uncertainty with</p><p>respect to the real model, and analyse the behaviour</p><p>of the total mass in the controlled system. It turns out</p><p>that, in this case, the asymptotical mass values lie in an</p><p>interval whose bounds can be easily expressed in terms</p><p>of the system uncertainties. Moreover, we prove that</p><p>for a class of compartmental systems in R3 of interest,</p><p>the state of the controlled system tends to an equilib-</p><p>rium point whose total mass lays within the aforemen-</p><p>tioned interval. Taking into account the relationship</p><p>between the mass and the state components in steady</p><p>state, it is possible to use the proposed mass control</p><p>law to track desired values for the steady state</p><p>components. This is applied to the control of the</p><p>neuromuscular blockade level of patients undergoing</p><p>surgery, by means of the infusion of atracurium.</p><p>Our results are illustrated by several simulations and</p><p>a clinical case.</p><p>*Corresponding author. Email: cnsousa@gaia.ipiaget.org</p><p>ISSN 00207179 print/ISSN 13665820 online</p><p> 2010 Taylor &amp; FrancisDOI: 10.1080/00207171003736311</p><p>http://www.informaworld.com</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Nor</p><p>th D</p><p>akot</p><p>a St</p><p>ate </p><p>Uni</p><p>vers</p><p>ity] </p><p>at 0</p><p>8:05</p><p> 14 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>2. Compartmental systems</p><p>Compartmental systems are dynamical systemsdescribed by a set of equations of the form</p><p>_xi Xj6i</p><p>fjix Xl6i</p><p>filx fi0x f0ix</p><p>i 1, . . . , n</p><p>(Sandberg 1978; Godfrey 1983) where x (x1, . . . , xn)Tis the state variable and xi and fi j take nonnegativevalues. Each equation describes the evolution of thequantity or concentration of material within a sub-system, called compartment. Since the compartmentsexchange matter with each other and with the envi-ronment, in the above equation, xi is the amount(or concentration) of material in compartment i, fi j isthe flow rate from compartment i to compartment jand the subscript 0 denotes the environment (Godfrey1983). In this article, we consider the class of lineartime-invariant compartmental systems described by</p><p>_xi Xj6i</p><p>kjixj Xl6i</p><p>kilxi qixi biu, i 1, . . . , n,</p><p>1</p><p>where xi and the input u take nonnegative values, therate constants, kij, as well as qi, bi are nonnegative andat least one bi is positive (Figure 1).</p><p>Note that, in this case, fji kjixj, f0i biu andfi0 qixi, and it can be easily proved that the systemverifies the following properties:</p><p>. it is positive, that is, if we consider an input uthat remains nonnegative, then the state var-iable also remains nonnegative;</p><p>. (1) can be written in matrix form as</p><p>_x Ax bu, 2</p><p>where A (called compartmental matrix) is suchthat</p><p>aii qi Xj 6i</p><p>kij and, if i 6 j, aij kji,</p><p>and b [b1 b2 . . . bn]T;. if u 0 and if the system is fully outflow</p><p>connected, (i.e. for every compartment i with</p><p>qi 0 there is a chain i! j! k! ! l withpositive rate constants at each step andwith ql40), then x 0 is a globally asymptot-ically stable equilibrium point of the system(Bastin and Provost 2002).</p><p>In this article, we shall only consider fully outflowconnected systems.</p><p>2.1 Mass control the exact case</p><p>The total mass of a compartmental system in a givenstate x is defined as Mx </p><p>Pni1 xi. For an arbitrary</p><p>positive value M , the set M fx 2 Rn :Mx M g of all the points x in the state spacewith mass M is called an iso-mass.</p><p>If u 0, the third property presented in the previ-ous section guarantees that the mass does not accu-mulate inside the system. However, this may nothappen if u is not always zero. This undesired situationis avoided if one is able to stabilise the total mass ofthe system in a given positive target value M , orequivalently, lead the state trajectories to the iso-massM .</p><p>This leads us to an important issue in the contextof the control of compartmental systems: to design acontrol law which yields a positive input that steers thesystem mass M(x) to a desired target value M .</p><p>In Bastin and Provost (2002), the positivecontrol law:</p><p>ux max 0, ~ux </p><p>~ux Xni1</p><p>bi</p><p> !1 Xni1</p><p>qixi M Mx !</p><p>,3</p><p>where is an arbitrary design parameter, is proposedfor this purpose and the desired convergence propertiesare obtained through the following result.</p><p>Theorem 2.1 (Bastin and Provost 2002): Let (2) be afully outflow connected compartmental system. Then, forthe closed-loop system (2)(3) with arbitrary initialconditions x0 2 Rn:</p><p>(i) the iso-mass M is forward invariant;(ii) the state x(t) is bounded for all t40 and</p><p>converges to the iso-mass M .</p><p>The proof of this theorem is based on the applica-tion of LaSalles invariance principle (LaSalle 1976,p. 30), by considering the Lyapunov function</p><p>Vx 12</p><p>M Mx 2</p><p>of (2) on Rn. We shall make use of an adapted versionof this function in our convergence analysis.</p><p>bi bj </p><p>qi qj</p><p>To/from other compartments</p><p>To/from other compartmentskji </p><p>kij i j</p><p>Figure 1. Two compartments of a linear time-invariantcompartmental model, as described by (1).</p><p>1388 C. Sousa et al.</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Nor</p><p>th D</p><p>akot</p><p>a St</p><p>ate </p><p>Uni</p><p>vers</p><p>ity] </p><p>at 0</p><p>8:05</p><p> 14 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>In Magalhaes et al. (2005), the control law (3) was</p><p>applied for the target control of the neuromuscular</p><p>blockade level of patients undergoing surgery, by</p><p>means of the infusion of atracurium. However, even</p><p>after a satisfactory identification of the patients</p><p>characteristics, it was necessary to consider an addi-</p><p>tional integrator, in order to achieve good results. This</p><p>might be explained by the fact that (contrary to what</p><p>happens, for instance, with state feedback stabilisers,</p><p>which are not uniquely defined from the system</p><p>matrices) the control law (3) strongly depends on the</p><p>system parameters. Since parameter uncertainty is</p><p>present not only in this case, but in most of the</p><p>applications, it is relevant to analyse the robustness of</p><p>that control law.</p><p>3. Mass control in uncertain compartmental systems</p><p>3.1 Robustness</p><p>We now prove that the control law presented in the</p><p>previous section leads the total mass of an uncertain</p><p>system to an interval whose bounds depend on the</p><p>parameter uncertainties.Since the control law (3) does not depend on the</p><p>interactions between compartments (i.e. it does not</p><p>depend on the kijs) and assuming that it is possible to</p><p>precisely measure what is injected from the outside</p><p>into the system (i.e. the parameters bi are not subject</p><p>to uncertainties), we shall consider the case where the</p><p>only uncertain parameters are q1, . . . , qn. Therefore, we</p><p>shall assume that a control law (3) is designed for a</p><p>nominal system</p><p>_x A DA x bu, 4</p><p>while the real system is given by</p><p>_x Ax bu, 5</p><p>and the matrix DA of parameter uncertainties isdiagonal. Moreover, we assume that the relative error</p><p>in the parameters is smaller than 100%, i.e.</p><p>DqiDAii is such that jDqij qi, i 1, . . . , n. Tuningthe control law (3) for this nominal system yields</p><p>ux max 0, ~ux </p><p>~ux Xni1</p><p>bi</p><p> !1 Xni1</p><p>qi Dqi xi M Mx !</p><p>:</p><p>6</p><p>It is proved in Sousa, Mendonca, and Rocha (2007)</p><p>that, for suitable values of the design parameter ,when the control law (6) is applied to (5), the</p><p>asymptotical values of the system mass lie in an</p><p>interval which is related to M as stated in the nexttheorem.</p><p>Theorem 3.1: Let (5) be a fully outflow connectedcompartmental system, Dqmax{jDqij} and take thedesign parameter in (6) larger than Dq. Then, the statetrajectories x(t) of the closed loop system (5)(6), witharbitrary initial conditions x0 2 Rn, converge to theforward invariant set</p><p> x 2 Rn : Mx 2 IM </p><p>,</p><p>with IM DqM , DqM h i</p><p>.</p><p>Remark 1:</p><p>(i) The set I(M ) is a neighbourhood of M . Thisbounds the absolute mass offset by</p><p>max M DqM</p><p>,</p><p>DqM M</p><p> DqDqM</p><p>,</p><p>leading to the bound DqDq for the relative massoffset. Clearly, this bound tends to zero whenDq goes to zero; this means that the controllaw is robust with respect to parameteruncertainty. Moreover, increasing the param-eter contributes to increasing the robustnessof the control law.</p><p>(ii) When Dqi 0, i 1, . . . , n, we recover theresult stated in Theorem 2.1.</p><p>(iii) Using the same kind of techniques as inTheorem 3.1, it is possible to show that,under the same conditions, the state trajecto-ries x(t) of the closed-loop system (5)(6), witharbitrary initial conditions x0 2 Rn, con-verge to the forward invariant set</p><p>~ x 2 Rn : Mx 2 ~IM </p><p>,</p><p>where ~IM Dqmin M, Dqmax M</p><p> andDqminmin{Dqi}, Dqmaxmax{Dqi}, if thedesign parameter is larger than Dqmax.This allows us to conclude that, in particular,if Dq1Dq2 DqnD, since DqminDqmaxD, the state trajectory x(t) convergesto the iso-mass M, with M DM .</p><p>(iv) Other bounds for the relative mass offset canbe derived from set ~IM ), namely</p><p>maxDqmin </p><p> Dqmin,</p><p>Dqmax </p><p> Dqmax</p><p> :</p><p>3.2 Mass control in neuromuscular relaxantadministration</p><p>In this subsection, some simulation examples arepresented for the control of the administration of the</p><p>International Journal of Control 1389</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Nor</p><p>th D</p><p>akot</p><p>a St</p><p>ate </p><p>Uni</p><p>vers</p><p>ity] </p><p>at 0</p><p>8:05</p><p> 14 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>neuromuscular relaxant drug atracurium to patients</p><p>undergoing surgery. It is possi...</p></li></ul>