Total mass control in uncertain compartmental systems

TOTAL MASS CONTROL IN UNCERTAIN COMPARTMENTAL SYSTEMS Claudia Sousa ∗,1Teresa Mendonca ∗∗ Paula Rocha ∗∗∗ ∗Escola Superior de Educacao Jean Piaget, V. N. Gaia, Portugal ∗∗ Faculdade de Ciencias da Universidade do Porto, Porto, Portugal ∗∗∗ Universidade de Aveiro, Aveiro Abstract: In this paper we analyse the control of the total mass of compartmental systems, under the presence of uncertainties. We consider a control law that has a very good performance when applied to compartmental systems without uncertainties and show that, even when the system parameters are not exactly known, that good performance is maintained. In fact, for a wide class of compartmental systems of R3, it is possible to prove that, when we apply that control law to the real system, the total mass of the system converges to a positive constant value, which depends on the parameter uncertainties and that can be made arbitrarily close to the desired mass, provided that the uncertainties in the parameter values are sufficiently small. The obtained results are illustrated by simulations for the control of the administration of the neuromuscular relaxant drug atracurium to patients undergoing surgery. Keywords: Compartmental systems, positive control, uncertain systems, neuromuscular blockade control, full outflow connectedness. 1. INTRODUCTION Compartmental models have been successfully used to model biomedical and pharmacokinetical systems, see, for instance, (Godfrey, 1983) or (Jacquez, 1993). This kind of systems consist of a finite number of subsystems, the compartments, which exchange matter with each other and with the environment. Such systems are positive systems (i.e., systems for which the state and output variables remain nonnegative whenever the input is nonnegative) and, as is well-known, in this case, the design of suitable control laws is more delicate, since one has to guarantee the positivity of the control input. In this framework, a nonnegative adaptive control law is proposed in (Haddad, 2003), in order to guarantee the partial asymptotic set-point stability of the closed loop system and a positive feedback control law is proposed in (Bastin, 2002), in order to stabilise the total system mass at an arbitrary 1This work was partially supported by FCT through the Unidade de Investigacao Matematica e Aplicacoes (UIMA), Universidade de Aveiro, Portugal. set-point. In (Magalhaes, 2005) the same positive control law proposed in (Bastin, 2002) was used for the control of the neuromuscular blockade (see (Lemos, 1991), (Linkens, 1994) and (Mendonca, 1998)) of patients undergoing surgery, but no analysis was made of the effect of parameter uncertainty in its performance. In this paper, we consider the control law proposed in (Bastin, 2002) and used in (Magalhaes, 2005), and analyse its performance for the control of the total mass of a wide class of compartmental systems, when the system parameters are not exactly known. Thus, we consider that the control law is tuned for a nominal process model that contains an additive uncertainty with respect to the real model, and analyse the behavior of the total mass in the controlled system. It turns out that, in this case, bounds for the asymptotical mass offset can be easily expressed in terms of the system uncertainties. Moreover, the total system mass converges to a positive constant value that depends on the value of the desired mass and on the uncertainties of the system parameters. 2. COMPARTMENTAL SYSTEMS Compartmental systems are dynamical systems described by a set of equations of the form ˙xi=X j6=i fji(x)−X l6=i fil(x)−fi0(x) + f0i(x) i= 1, . . . , n (see (Godfrey, 1983) or (Sandberg, 1978)) where x= (x1, . . . , xn)Tis the state variable and xiand fij take nonnegative values. Each equation describes the evolution of the quantity or concentration of material within a subsystem, called compartment. Since the compartments exchange with each other and with the environment, in the above equation, xiis the amount (or concentration) of material in compartment i,fij is the flow rate from compartment jto compartment iand the subscript 0denotes the environment (see (Godfrey, 1983)). In this paper, we consider the class of linear time-invariant compartmental systems described by ˙xi=X j6=i kjixj−X l6=i kilxi−qixi+biu, (1) i= 1, . . . , n, where xiand the input utake nonnegative values, kij, qi, bi∈R+and at least one biis positive (see Fig. 1). Note that, in this case, fji =kjixj, f0i=biuand fi0=qixi, and it can be easily proved that the system is positive, this is, if we consider an input uthat remains nonnegative, then the state variable also remains nonnegative. Moreover, (1) can also be written in matrix form as ˙x=Ax +bu, (2) where A(called compartmental matrix) is so that aii =−qi−X j6=i kij and, if i6=j, aij =kji, and b= (b1, b2, . . . , bn)T. 2.1 Stabilisation The total mass of the system in a given state xis defined as M(x) = Pn i=1 xi. For an arbitrary positive b i b j q i q j to/from other compartments to/from other compartments k j i k i j i j Fig. 1. Two compartments of a linear time-invariant compartmental model, as described by (1). value M∗,the set ΩM∗=©x∈Rn +:M(x) = M∗ª of all the points xin the state space with mass M∗is called an iso-mass. An important issue in the context of the control of compartmental systems that is to design a control law which yields a positive input that steers the system mass M(x)to a desired value. In (?), the following positive control law is proposed in order to guide the system trajectories to a given isomass ΩM∗: u(x) = max (0,˜u(x)) ˜u(x) = Ãn X i=1 bi!−1 ((M(x))) Ãn X i=1 qixi+ +λ(M∗−M(x))) , (3) where λis an arbitrary design parameter. In order to state the corresponding theorem, we need to introduce the following concept of full outflow connectedness.A compartmental system (1) is said to be fully outflow connected if at every state xthere is a path i→j→k→ · · · → lwith positive kij’s from every compartment ito some compartment lsuch that ql>0(see (?)). Theorem 1. (Bastin, 2002) Let (2) be a fully outflow connected compartmental system. Then, for the closed loop system (2)-(3) with arbitrary initial conditions x(0) ∈Rn +: i)the iso-mass ΩM∗is forward invariant; ii)the state x(t)is bounded for all t > 0and converges to the iso-mass ΩM∗. In (Magalhaes, 2005), the control law (3) was applied for the target control of the neuromuscular blockade of patients undergoing surgery, by means of the infusion of atracurium. However, even after a satisfactory identification of the patients characteristics, it was necessary to consider an additional integrator, in order to achieve good results. This might be explained by the fact that (contrary to what happens, for instance, with state feedback stabilisers, which are not uniquely defined from the system matrices) the control law (3) strongly depends on the system parameters. Since parameter uncertainty is present not only in this case, but in most of the applications, it is relevant to analyse the robustness of that control law. 3. MASS CONTROL UNDER UNCERTAINTY In this section, we analyse the performance of the control law (3), proposed in (Bastin, 2002), in the presence of parameter uncertainties. If we consider that we can precisely measure what is injected from the outside into the system, the parameters biare not subject to uncertainties. On the other hand, since the control law does not depend on the interactions between compartments, that is, it does not depend on the kij’s , we may assume that the only uncertain parameters are q1, . . . , qn. Therefore, we shall assume that a control law (3) is designed for a nominal system ˙x= (A+ ∆A)x+bu, (4) while the real system is given by ˙x=Ax +bu, (5) being ∆Athe matrix of parameter uncertainties. Note that, in this case, if ∆qi=−∆Aii, i = 1, ..., n, the control law (3) becomes u(x) = max (0,˜u(x)) ˜u(x) = Ãn X i=1 bi!−1Ãn X i=1 (qi+ ∆qi)xi+ +λ(M∗−M(x))) . (6) It turns out that, for suitable values of the design parameter λ, when the control law (6) is applied to (5), the asymptotical values of the system mass lay in an interval which is related to M∗as stated in the next theorem. Theorem 2. (Sousa, 2007) Let (5) be a fully outflow connected compartmental system, ∆qmin = min {∆qi},∆qmax = max {∆qi}and take the design parameter λin (6) larger than ∆qmax.Then, the state trajectories x(t)of the closed loop system (5)-(6), with arbitrary initial conditions x(0) ∈Rn +, converge to the forward invariant set Ω = ©x∈Rn +:M(x)∈I(M∗)ª, with I(M∗) = hλ λ−∆qmin M∗,λ λ−∆qmax M∗i. The previous result, gives us sufficient conditions to generalize Theorem 1 for a wide class of uncertain compartmental systems of R3. Indeed, if we consider fully outflow connected compartmental systems for which qi6= 0, i = 1,2,3and qmin + ∆qmin > ∆qmax(which is a reasonable assumption for practical issues), it is possible to prove that, when the control law (6) is applied to (5), the system mass converges to a constant value in the aforementioned interval, which is related to M∗as stated in the next results. Proposition 3. Consider that qmin+∆qmin >∆qmax. Take the design parameter λin (6) larger than ∆qmax. Then, when the control law (6) is applied to (5), there exists an instant t1>0such that, for t≥t1, u(x(t)) = eu(x(t)) ≥0, PROOF. According to 2, when the control law (6) is applied to (5), the asymptotical values of the system mass lay in the interval I(M∗) = £Mmin, Mmax¤, providing the design parameter λin (6) to be larger than ∆qmax.This implies that, for every ε > 0, there exists an instant t1>0such that M(x(t)) ∈£Mmin −ε, Mmax +ε¤, for t≥t1. Note that ˜u(x)≥0⇔ ⇔ 3 X i=1 (qi+ ∆qi)xi+λ(M∗−M(x)) ≥0 ⇔ n X i=1 (λ−qi−∆qi)xi≤λM∗. Since, for t≥t1, n X i=1 (λ−qi−∆qi)xi(t)≤ ≤(λ−qmin −∆qmin)¡Mmax +ε¢, and (λ−qmin −∆qmin)¡Mmax +ε¢≤λM∗⇔ ⇔ε≤λ λ−qmin −∆qmin M∗−Mmax ⇔ε≤λ λ−qmin −∆qmin M∗−λ λ−∆qmax M∗, if we take ε1=λ λ−qmin−∆qmin M∗−λ λ−∆qmax M∗it follows that, if ε≤ε1,˜u(x(t)) ≥0. Thus, considering t1>0such that, for t≥t1, M(x(t)) ∈£Mmin −ε1, Mmax +ε1¤, we guarantee that, for t≥t1,u(x) = ˜u(x)≥0.Note that, as qmin + ∆qmin >∆qmax,ε1>0.2 In the following, we will only consider compartmental systems of R3. Take k=¡P3 i=1 bi¢−1and consider the following points: ¯x1= [α1¯x13 α2¯x13 ¯x13]T, with α1=k21 +b1kq2 q1+k12 +k13 −b1kq1 α2+ +k31 +b1kq3 q1+k12 +k13 −b1kq1 α2=b2k(q1q3+q3k13 +q1k31) α3 + +(b2k+b1k)q3k12 α3 + +k32 [q1(1 −b1k) + k12 +k13] + k12k31 α3 α3= (1 −b1k−b2k) (q2q1+q2k12 +k21q1) + +q2k13 +k21k13 +k23 [q1(1 −b1k) + k12 +k13] ¯x13 =λM∗ (λ−∆q1)α1+ (λ−∆q2)α2+ (λ−∆q3). ¯x2= [α1¯x22 ¯x22 0]T, with α1=k21 +b1kq2 q1+k12 −b1kq1 ¯x22 =λM∗ (λ−∆q1)α1+ (λ−∆q2) and ¯x3=·λM∗ λ−∆q1 0 0¸T . Proposition 4. If qi6= 0, i = 1,2,3,qmin +∆qmin > ∆qmax and if the design parameter λin (6) is larger than ∆qmax = max {∆qi}, the closed loop system (5)-(6) has a unique equilibrium point that belongs to the set nx∈R3 +:dM(x) dt = 0o.If the third component of b= [b1b2b3]Tis positive or if it is zero but there is something coming into that compartment, then, the equilibrium point is of the form ¯x1. If the third component of b= [b1b2b3]Tis zero and k13 = k23 = 0, that is, if there is nothing coming into that compartment, then, there are two cases to be considered: •the equilibrium point is of the form ¯x2if the second component of bis positive or if it is zero but there is something coming into that compartment, that is, k12 6= 0 •the equilibrium point is of the form ¯x3if the second component of bis zero and there is nothing coming into that compartment, that is, k12 = 0. 2 PROOF. According to Proposition 3, there exists an instant t1>0such that, for t≥t1,u(x(t)) = ˜u(x(t)) ≥0.Thus, for t≥t1, dM(x(t)) dt = 0 ⇔ ⇔ 3 X i=1 ∆qixi(t) + λ(M∗−M(x(t))) = 0 ⇔(λ−∆q1)x1(t)+(λ−∆q2)x2(t)+ + (λ−∆q3)x3(t) = λM∗ and the equilibrium points of (5)-(6) that belong to the set nx∈R3 +:dM(x) dt = 0oare the solutions of the system (Ax +bu = 0 dM(x) dt = 0 .(7) But, in this case, u(x) = ˜u(x) = =Ã3 X i=1 bi!−1Ã3 X i=1 (qi+ ∆qi)xi+λ(M∗− −M(x))) =Ã3 X i=1 bi!−1Ã3 X i=1 qixi! =Ã3 X i=1 bi!−1 [q1q2q3]x, thus, (7) becomes             A+bÃ3 X i=1 bi!−1 [q1q2q3] x= 0 (λ−∆q1)x1(t)+(λ−∆q2)x2(t)+ + (λ−∆q3)x3(t) = λM∗ . and it is easy to verify that, for each mentioned case, x1,x2and x3are the only solutions of this system. 2 Theorem 5. Consider that qi6= 0, i = 1,2,3and qmin + ∆qmin >∆qmax. For each situation mentioned in Proposition 4, the state trajectories x(t)of the closed loop system (5)-(6), with arbitrary initial conditions x(0) ∈R3 +, converge to the corresponding equilibrium point x1,x2, or x3, providing the design parameter λin (6) to be larger than ∆qmax = max {∆qi}. Corollary 6. Consider that qi6= 0, i = 1,2,3and qmin + ∆qmin >∆qmax and let M1,M2and M3 be the total mass of the system in ¯x1,¯x2and ¯x3, respectively. Then, the state trajectories x(t)of the closed loop system (5)-(6), with arbitrary initial conditions x(0) ∈ Rn +, converge to ©x∈R3 +:M(x) = M1ª,©x∈R3 +: M(x) = M2ªor to ©x∈R3 +:M(x) = M3ª, providing the design parameter λin (6) to be larger than ∆qmax = max {∆qi}. PROOF. It is only necessary to note that M1is the total mass of the system in ¯x1,M2is the total mass of the system in ¯x2and M3is the total system mass of the system in ¯x3. The convergence to the aforementioned iso-masses follows from Theorem 5. 2 In the following, we prove Theorem 5. PROOF. Suppose that the third component of b= [b1b2b3]Tis positive or that it is zero but there is something coming into that compartment. Let ¯x1be the equilibrium point mentioned in (4), M1=(α1+α2+ 1) λM∗ (λ−∆q1)α1+ (λ−∆q2)α2+ (λ−∆q3) be the total mass of the system in ¯x1and take k= ¡P3 i=1 bi¢−1. Take t1such that, for t≥t1,u(x(t)) = ˜u(x(t)) ≥0 (the existence of such an instant is guaranteed by (3)). Since M∗= =(λ−∆q1)α1+ (λ−∆q2)α2+ (λ−∆q3) (α1+α2+ 1) λM1 =M1−∆q1α1+ ∆q2α2+ ∆q3 (α1+α2+ 1) λM1, for t≥t1 u(x(t)) = ˜u(x(t)) = =k"3 X i=1 (qi+ ∆qi)xi(t) + λ(M∗−M(x(t)))# =k"3 X i=1 qixi(t) + n X i=1 ∆qi(xi(t)−¯x1i) + +λ¡M1−M(x(t))¢¤ =k[[q1q2q3]x(t) + [∆q1∆q2∆q3] (x(t)−¯x1) + +λ[1 1 1] ( ¯x1−x(t))] and ˙ z }| { (x(t)−¯x1) = =Ax(t) + bu −(A+bk [q1q2q3]) ¯x1 =Ax(t) + b[k[q1q2q3]x(t)+ +k[∆q1∆q2∆q3] (x(t)−¯x1) + +kλ [1 1 1] ( ¯x1−x(t))] −(A+bk [q1q2q3]) ¯x1 = (A+bk [q1q2q3] + bk [∆q1∆q2∆q3]− −bkλ [1 1 1]) (x(t)−¯x1) =¯ A(x−¯ x1) (t). Since it can be easily seen that, for λ > ∆qmax, all the eigenvalues of ¯ Alie in C-, it turns out that ¯ Ais asymptotically stable and hence (x−¯x1) (t)→0or, equivalently, x(t)→¯x1. To prove the convergence to ¯x2or to ¯x3, we only need to consider the total mass of the system in ¯x2and in ¯x3, that is, M2=(α1+ 1) λM∗ (λ−∆q1)α1+ (λ−∆q2) and M3=λM∗ λ−∆q1 and the proof will follow as in the previous case. 2 Note that, in this case, we may bound the absolute mass offset by |∆qmax|M∗ λ−∆qmin and, clearly, it tends to zero when the uncertainties go to zero; moreover, increasing the design parameter λcontributes to increase the robustness of the control law. 3.1 Case Study - mass control in neuromuscular relaxant administration In this section, some simulation examples are presented for the control of the administration of the neuromuscular relaxant drug atracurium to patients undergoing surgery. It is possible to model this problem as a three compartmental model that can be described as depicted in Fig. 2, where uis the drug infusion dose administered in the central compartment, and k12, k21, k13, q3are positive micro-rate constants and q1, q2are nonnegative micro-rate constants that vary from patient to patient. k 1 3 q 1 q 2 k 2 1 k 1 2 Central Compartment (C1) Peripheral Compartment (C2) b 1 =1 Effect Compartment (C3) q 3 Fig. 2. Compartmental model for the neuromuscular blockade effect of the drug atracurium. We consider the following values for the parameters (units =min−1): k12 = 0.1928, k13 = 0.0017, k21 = 0.1556, q1= 0.1047, q2= 0.1, q3= 0.0836. 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Mass (µg/kg) Time (min) system mass desired mass asymptotical mass Fig. 3. Simulation for the neuromuscular blockade control, considering ∆qi= 0, i = 1,2,3. (a) 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (min) Mass (µg/kg) system mass desired mass asymptotical mass (b) 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Mass (µg/kg) Time (min) system mass desired mass asymptotical mass (c) 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Mass (µg/kg) Time (min) system mass desired mass asymptotical mass Fig. 4. Simulations for the neuromuscular blockade control. (a) Simulation with ∆qi= 0.03, i = 1,2,3.; the system mass reaches the set-point M1= 84.7662. (b) Simulation with ∆q1=−0.09,∆q2=−0.03 and ∆q3= −0.02; the system mass reaches the set-point M1= 52.2288. (c) Simulation with ∆q1= 0.02,∆q2= 0 and ∆q3= 0.05; the system mass reaches the set-point M1= 78.3715. Our aim is to stabilise the system mass on the value M∗= 72.0513 (which, in an exact modelling situation, would correspond to the typical 10% level of neuromuscular blockade), using the control law (6). We start by taking the design parameter λ= 0.2. In the first simulation, depicted in Fig. 3, it is assumed that the nominal patient model coincides with the real one, i.e., ∆qi= 0, i = 1,2,3. As expected, the system mass converges to M∗. The first simulation in Fig. 4 corresponds to the case where the ∆qi’s are taken to be all equal, namely ∆q1= ∆q2= ∆q3= 0.03. The second and the third simulations in Fig. 4 correspond to the case where the ∆qi’s are different. Finally, Fig. 5 illustrates the behavior of the mass of the controlled system for different values of the parameter λ, under a fixed uncertainty for the system parameters. According to the definition of M1, one observes that the increasing of λcorresponds to the decrease of the final mass offset. (a) 0 50 100 150 200 250 300 0 20 40 60 80 100 120 Mass (µg/kg) Time (min) system mass desired mass asymptotical mass (b) 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Mass (µg/kg) Time (min) system mass desired mass asymptotical mass (c) 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Mass (µg/kg) Time (min) system mass desired mass asymptotical mass Fig. 5. Simulations for the neuromuscular blockade control. These simulations where obtained considering ∆q1=−0.02,∆q2= 0 and ∆q3= −0.03, and different values of λ. (a) Simulation for λ= 0.02. (b) Simulation for λ= 0.2. (c) Simulation for λ= 2. 4. CONCLUSION This paper presents a study of the performance of the control law (3) proposed in (Bastin, 2002) when applied to the mass control of compartmental systems with parameters uncertainties. It turns out that the asymptotical mass values converge to a positive constant value that can be expressed in terms of the parameter uncertainties and of the desired mass. 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