Observer-Based Output Feedback Linearization Control with

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Observer-Based Output Feedback Linearization Control with Application to HIV Dynamics Iman Hajizadeh, and Mohammad Shahrokhi Ind. Eng. Chem. Res., Just Accepted Manuscript • Publication Date (Web): 30 Jan 2015 Downloaded from http://pubs.acs.org on February 3, 2015

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Industrial & Engineering Chemistry Research

Figure 1. a) Evolution of the viral load, b) evolution of the CD4+T cells concentration, c) RTI efficacy, d) PI efficacy in case of model parameter uncertainty applied at t=100 day. 127x100mm (600 x 600 DPI)

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Figure 2. a) Evolution of the viral load, b) evolution of the CD4+T cells concentration, c) RTI efficacy, d) PI efficacy for noise corrupted measurements. 127x100mm (600 x 600 DPI)


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Figure 3. a) Evolution of the viral load, b) evolution of the CD4+T cells concentration, c) RTI efficacy, d) PI efficacy using the LNO observer. 127x100mm (600 x 600 DPI)



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Figure 4. a) Evolution of the viral load, b) evolution of the CD4+T cells concentration, c) RTI efficacy d) PI efficacy in case of sparse measurements. 127x100mm (600 x 600 DPI)


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Figure 5. a) Evolution of the viral load, b) evolution of the CD4+T cells concentration, c) RDV dose, d) 3TC dose for impulsive control strategy. 127x100mm (600 x 600 DPI)



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Observer-Based Output Feedback Linearization Control with Application to HIV Dynamics Iman Hajizadeh, Mohammad Shahrokhi* Department of Chemical and Petroleum Engineering, Sharif University of Technology, P.O. Box 11155-9465 Azadi Av., Tehran, Iran

ABSTRACT: This paper presents the feedback linearization control of HIV infection. A multiinput multi-output (MIMO) dynamic nonlinear HIV infection model for this purpose has been used. For this purpose, three widely used drugs are considered. A Luenberger-like nonlinear observer (LNO) is designed for estimation of unavailable states. To minimize the side effects of drugs, the concentration of ZDV which has the highest side effect is fixed to a minimum value and the external controllers parameters are obtained by maximizing an objective function. In the control design, limitations on drugs consumption and unavailability of all states are taken into account. The closed- loop stability has been established in the presence of the observer dynamics and input limitations. Robustness of the proposed treatment strategy against model parameter uncertainties, noise or sparse measurements has been demonstrated through simulation study. Finally, by including pharmacological models, an impulsive control strategy for HIV treatment has been considered. 1 Environment ACS Paragon Plus

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1. INTRODUCTION The HIV infection is one of the greatest world health problems. In 2009, the number of infected people with the HIV has been estimated to be 33.3 million and 1.8 million have died because of HIV infection.1 Due to this infection, the vital human cells such as CD4+T-cells become infected which results in destruction of human immune system. Under this condition, the probability that human body becomes infected by the opportunistic infections increases drastically.2 In recent years, the HIV treatment has been improved significantly and has reached to an efficient treatment approach called highly active antiretroviral therapy (HAART). In the common HIV treatment, patients are recommended to take several antiretroviral drugs from two different classes every day. The common anti-HIV drugs consist of two categories, namely reverse transcriptase inhibitors (RTIs) and protease inhibitors (PIs). In the HAART usually two or more drugs, typically one or more from RTI type and one from PI type, are used. Because of drug side effects, continuous HIV treatment maybe infeasible and treatment should be stopped during the first year. Due to the above limitation, finding an efficient drug regimen has become a challenging problem.3, 4 Using non-linear differential equations, the HIV infection dynamics can be modeled. Several mathematical models have been developed in the literature and most of them describe the interaction of the HIV with the CD4+ T cells.5-17 Having the HIV infection dynamics, different control approaches can be used for reducing the viral load and increasing the healthy white blood cells (T-cells). For instance, optimal control,3, 12, 18-31

model predictive control (MPC),32-37 global linearizing control (GLC),4, 38-41 sliding mode

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control,42 and fuzzy control43-45 have been examined in many research works. In what follows, some of these works are reviewed briefly. Culshaw, et al. 28 presented an optimal control scheme based on the action of RTIs in order to maximize the healthy CD4+ T cells and immune response cells. Costanza, et al.

3

designed a

treatment policy by defining an objective function and using the optimal control strategy. They presented both open-loop and closed-loop results of their approach based on the action of RTIs. Mhawej, et al. 4 used input output feedback linearization control approach and incorporated the pharmacokinetics (PK) and pharmacodynamics (PD) of Zidovudine to design a controller for obtaining a suitable dosage regimen. Rivadeneira and Moog

38

designed an impulsive control

strategy based on the exact feedback linearization technique by taking into account the PK and PD of Zidovudine. Barão and Lemos

39

used a singular perturbation approximation to simplify

the HIV model and based on the reduced model have designed a feedback linearizing controller. Pinheiro and Lemos

35

proposed a nonlinear model predictive control (NMPC) algorithm

embedding the nonlinear multirate state estimation via an extended Kalman filter. Their model had two inputs representing the efficacies of RTIs and PIs drugs. Pinheiro, et al.

36

proposed a

periodic nonlinear MPC, using a model incorporating drugs PK and PD, to compute optimal drug dosage for HIV infection treatment. Radisavljevic-Gajic

27

proposed a control strategy based on

linearization of a nonlinear model of HIV infection at the equilibrium point based on the actions of RTIs and PIs. Zarrabi, et al. 42 used the sliding mode control to design a controller for the HIV infection treatment based on application of RTI drugs. Assawinchaichote and Junhom

45

presented a design of H∞ fuzzy controller for the HIV infection based on the actions of RTIs and PIs.


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In designing the HIV treatment strategy two constraints, namely unavailability of all states and drug consumption limitation, should be considered. To solve the first problem, a state observer can be incorporated in the control design.21, 35-37, 46 To handle the second problem, input (drug) limitations should be considered as a constraint in designing the controller.20, 30 To the best of authors’ knowledge, a control strategy which considers these two limitations in the control design and its closed-loop stability has been established is not proposed for the HIV treatment in the literature. In this paper a MIMO feedback linearization strategy is utilized for designing a control scheme for the HIV infection treatment. In order to design an effective control system, an appropriate model whose parameters are selected based on clinical data is used.47 In this model, the system outputs are the total concentration of CD4+T cells and the viral load.4,

21, 26, 46

Since for

implementation of the control scheme system states are required, a Luenberger-like nonlinear observer (LNO) is designed for estimation of unmeasurable states. Combinations of three widely used drugs are considered for the treatment and the side effects of drugs are minimized by applying an optimization strategy.18, 20 In designing the controller, drug consumption limitation has been taking into account and the stability of the closed-loop system in presence of the observer dynamics and input limitation has been established. Effects of model parameter uncertainties, noise or sparse measurements are investigated through simulation study. Finally PK and PD models have been included in the HIV infection model to design a treatment strategy based on consumption of three widely used drugs. The paper is organized as follows. In section 2, the HIV infection model is described. Control and observer designs are presented in section 3. In section 4 by using the pharmacological concepts, drug efficacies are related to drugs plasma concentrations and controller parameters



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have been tuned. The results and discussions are given in section 5. The conclusions are drawn in section 6. 2. HIV INFECTION DYNAMIC MODEL Although the HIV infection dynamics are very complex and involve multiple interactions between the virus and the host immune system, but the main characteristics of this infection can be determined from the relatively simple equations. In this study the following model which characterizes the viral dynamics for a patient has been used:47  x& (t ) = λ − d x (t ) − β (1 − u 1 ) x (t )v (t )   y& (t ) = β (1 − u 1 ) x (t )v (t ) − a y (t ) + λ y & v (t ) = γ (1 − u 2 ) y (t ) − ω v (t )

(1)

where x is the concentration of target CD4+T cells; y is the concentration of actively infected CD4+T cells; v is the viral load; λ is the proliferation rate; d is the death rate of target CD4+T cells; β is the infection rate; a is the death rate of actively infected cells; λy is the contribution of the reservoir to actively infected CD4+T cells; γ is the rate of free virus production by infected cells; ω is the clearance rate for the free virus; u1 is the effect of RTI drugs and u2 is the effect of PI drugs. The model parameters are taken from the Luo, et al.

47

paper which are the average values of

the parameters obtained from the clinical data of ten patients. For each parameter, the mean value is given below: λ = 292 cells µL-1 day-1; d = 0.1825 day-1; β = 3.9×10-6 ml copies-1 day-1; a = 1.3 day-1; λy= 0.0004 cells µL-1 day-1; γ = 5935 copies µLcells-1 ml-1 day-1; ω = 18.8 day-1.


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3. FEEDBACK LINEARIZATION CONTROL OF HIV INFECTION One of the common nonlinear control approaches is the feedback linearization control strategy.48 First a brief introduction of this technique is presented. Consider the following MIMO nonlinear system: m

X& ( t ) = f ( X ) + ∑ g i ( X )u i

(2)

i =1

y i = hi ( X ),

i = 1, 2 ,..., m

where X is an n-dimensional state vector and u and y are m-dimensional input and output vectors. The input output relation can be obtained by differentiating the output for r times. The number r is called the relative degree of the system. The MIMO system has a vector relative degree

[ r1 , r2 … rm ] at a point X0 if

L g j Lkf hi (X ) = 0 for all 1 ≤ j ≤ m, all 1 ≤ i ≤ m, k 0, L ( X ) − L ( Xˆ ) ≤ γ L X%

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(14)

where X% = X − Xˆ .

Assumption 3. The control inputs are bounded, i.e.,

∀t ≥ 0, 0 ≤ u1,min ≤ u1 ≤ u1,max

(15)

∀t ≥ 0, 0 ≤ u2,min ≤ u2 ≤ u2,max

(16)

Assumption 4. For a given matrix kLNO, the positive parameter q exists which satisfies the following inequality:

(−q + 2 Pm γ φ−1 ((β (d − a) γ L + d 2 + a2 ) + (γ + ω) + (u1,max β (d − a) γ L ) + (u 2,max γ ))) < 0

(17)

where Pm is the maximum eigenvalue of a positive definite matrix P. To check the assumption 1, the rank of observability matrix should be calculated. It has been shown in the appendix B that observability matrix is full rank and therefore system (7) is observable. The observer equation is as follows: & ′ ( y − h ( Xˆ )) Xˆ = f ( Xˆ ) + g 1 ( Xˆ ) u 1 + g 2 ( Xˆ ) u 2 + k LNO  yˆ 1   xˆ + yˆ   h1 ( Xˆ )    yˆ  =  vˆ  =    h2 ( Xˆ )   2 

(18)

where

λ − d xˆ (t ) − β xˆ (t ) vˆ (t )   β xˆ (t ) vˆ (t )   0        ˆ ˆ ˆ f (X ) =  β xˆ (t ) vˆ(t ) − a yˆ (t ) + λy  , g 1 ( X ) =  − β xˆ (t ) vˆ (t )  , g 2 (X ) =  0  γ yˆ (t ) − ω vˆ (t )     ˆ 0    − γ y (t )   

′ , k LNO

 1 1 0 −1 ˆ ˆ = O (X )k LNO , O (X ) =  −d − a 0  and Xˆ = [xˆ , yˆ ,vˆ]T .  0 0 1 


(19)

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The kLNO matrix is determined based on a pole placement approach to assign the desired observer poles. Using the above assumptions, it can be shown that the estimation error goes to zero as t → ∞.50, 51

3.2. COUPLING THE OBSERVER WITH THE FEEDBACK CONTROLLER For implementing the control law (20) in the absence of state measurements, the following control law can be used.

ud = −aˆ−1(Xˆ )bˆ(Xˆ ) + aˆ−1(Xˆ )υ

(21)

where  L gˆ L1ˆ hˆ1 ( Xˆ ) L gˆ L1ˆ hˆ1 ( Xˆ )   β xv ˆ ˆ (a − d ) 0 f f 1 2 =  aˆ ( Xˆ ) =  ˆ ˆ − γ yˆ   L gˆ 2 hˆ2 ( Xˆ )   0  L gˆ1 h 2 ( X )

(22)

 L 2ˆ hˆ1 ( Xˆ )   − (d + k 1,′ LNO + k 2,′ LNO ) fˆ1 − (a + k 1,′ LNO + k 2,′ LNO ) fˆ2  f ˆ ˆ =  b (X ) =  1 ˆ ˆ ( Xˆ ))  L ˆ h 2 ( Xˆ )    ′ ˆ ˆ y − v + k y − h γ ω ( 3, LNO 2 2  f  

(23)

fˆ1 = λ − d xˆ (t ) − β xˆ (t ) vˆ (t ) + k 1,′ LNO ( y 1 − h1 (Xˆ ))

(24)

and

fˆ2 = β xˆ (t ) vˆ(t ) − a yˆ (t ) + λy + k 2,′ LNO ( y 1 − h1 (Xˆ ))

(25)

Using equation (21), the desired individual control inputs are:

ud ,1 =

υ1 − L2fˆ hˆ1 (Xˆ )

(26)

Lgˆ1 L1fˆ hˆ1 (Xˆ )

and



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u d ,2 =

υ 2 − L1fˆ hˆ2 (Xˆ )

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(27)

L gˆ 2 hˆ2 ( Xˆ )

υ1 and υ2 are given by the following equations:

υ1 = −k 1 e1 − k 2 e2 − k 3 e3 + y&&1,d +υ11

(28)

and

υ2 = −k 4 e4 − k 5 e5 + y& 2,d

(29)

where

e1 = (yˆ1 − y d ,1) d t  ∫ e2 = yˆ1 − y d ,1  & e3 = yˆ1 − y&d ,1  e4 = ∫ (yˆ2 − y d ,2 ) d t  e5 = yˆ2 − y d ,2

(30)

In equation (28), the υ11 term is added to cope with the error caused by state estimates. In the above equations, yd,1 and yd,2 are the desired outputs. If υ11 is chosen as given below and the system is bounded input bounded output stable (BIBO), the asymptotic closed loop stability including the observer dynamics can be established:

υ11 = (k 1,′ LNO + k 2,′ LNO )(λ + λy )(d xˆ + a yˆ )

(31)

k1, k2, k3, k4 and k5 are selected such that all roots of the two polynomial s3+k3s2+k2s+k1 and s2+k5s+k4 lie in the open left-hand side of the complex plane. It should be noted that the BIBO assumption is valid for the HIV dynamics. The asymptotic stability of the proposed control strategy for HIV treatment in the presence of observer dynamics and input limitation is established in the appendix A.


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4. CONTROLLERS TUNING USING THE PHARMACOLOGICAL CONCEPTS The current section addresses the relation between the drug efficacies to the drug plasma concentrations by incorporating the pharmacological models. The drug efficacies are related to drug concentrations by the following equations:20 C 2 (t ) C 3 (t ) + IC 502 IC 503 u1 (t ) = C (t ) C (t ) 1+ 2 2 + 3 3 IC 50 IC 50

(32)

and

u 2 (t ) =

C 1 (t ) IC + C 1 (t )

(33)

1 50

1 2 3 where u1 and u2 are the RTI and PI efficacies, and C1, IC 50 , C2, IC 50 , and C3, IC 50 show the

plasma and median inhibitory concentrations of RDV, 3TC, and ZDV, respectively. One of the vital factors that must be considered in the standard HIV treatment is minimizing the drug side effects. Since ZDV (C3) has a greater side effect than the other two drugs, its concentration is fixed to a minimum value.20 By fixing the concentration of C3, concentrations of the remaining drugs are related to drug efficacies via the following equations:

C 3 (t ) = C 3,min

C 2 (t ) = IC 502

where κ =

(34)

(1 − κ ) u1 (t ) − κ 1 − u1 (t )

C 3,min IC 503

1 C 1 (t ) = IC 50

(35)

.

u 2 (t ) 1 − u 2 (t )

(36)



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The parameters of the external controllers (k1, k2, k3, k4, and k5) can be obtained by maximizing the following objective function: tf

J = ∫ [ A1 x − A 2 S e ] dt

(37)

t0

where x is the concentration of target CD4+T cells, Se represents the total side-effects of drugs which is a function of drug concentrations, and A1 and A2 are weighting factors which have been set to 1 mm3 and 1000, respectively.18, 20 In addition minimum and maximum values were considered for each drug concentration to ensure to follow the antiretroviral guidelines for the HIV treatment. In the present study, the following values which lie in the standard range of drug dosages are used for minimum and maximum drug concentrations:20

C 1,min = 0.0122 mg L-1 , C 1,max = 0.081 mg L-1 , C 2,min = 0.0205 mg L-1 , C 2,max = 0.4608 mg L-1 , and C 3,min = 0.00659 mg L-1 . The maximum and minimum values of efficacies are obtained by using the above values in equations (32) and (33) which are u1,min= 0.1, u1,max= 0.7105, u2,min= 0.1, and u2,max= 0.4241, respectively. To obtain the optimal values of controller parameters, for a selected set of controller parameters, the efficacies are evaluated for the period of treatment. Using equations (35) and (36), C1 and C2 are obtained. Since C3 is fixed, Se which is function of drug concentrations can be calculated. Having Se, the performance index (37) can be evaluated. The above procedure is repeated until this performance index is maximized for the selected controller parameters. The pattern search technique has been used for solving the above optimization problem.


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5. RESULTS AND DISCUSSION In this section the performance of the proposed control scheme will be investigated under different conditions. The main control objectives are:27, 38

•

Decreasing the viral load to 10% of its initial value within eight weeks.

•

Decreasing the viral load to less than 50 copies ml-1 within six months of the treatment.

•

Increasing the CD4+T cells concentration to a vicinity of its normal value ( λ / d = 1600 cells µL-1 ).

The desired steady state values of yd,1 and yd,2 are considered as: yd,1= 1599 cells µL-1 and yd,2= 30 copies ml-1. The initial conditions for the patient states are set to:

x ( 0 ) = 1056 cells µL-1 , y ( 0 ) = 76 cells µL-1 , v ( 0 ) = 24115 copies ml-1 To improve the performance of the external controllers, the anti-wind-up scheme proposed by Yoon, et al. 52 has been used.

5.1 CONTROLLER PERFORMANCE IN PRESENCE OF MODEL PARAMETER UNCERTAINTIES In this section the results of implementation of the feedback linearizing control strategy in the presence of model parameter uncertainties are presented. Controllers gains are obtained by maximizing the performance index (37), using the pattern search technique and given below: k 1 = 0.0218 , k 2 = 0.2340 , k 3 = 0.8379 , k 4 = 4.3631 , k 5 = 4.1776 It is desired to change the patient state from the initial condition to the desired status. To check the robustness of the proposed control strategy against model mismatch, at t=100 day the γ



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coefficient in the model has been increased by 20 %. The simulation results are shown in Fig. 1. As can be seen the viral load drops under 50 copies/ml and the CD4+T cells concentration increases from its initial value to its desired value in less than 60 days (Fig. 1 (a) and (b)). Thus, the designed control law fulfills the desired objectives. Regarding model parameter uncertainty, it is observed that the controller is able to compensate the effect of model mismatch. The related drug efficacies are shown in Fig. 1 (c) and (d). The maximum model parameter uncertainties rejected by the proposed controller are given in Table 1.

Figure 1 Table 1 5.2 CONTROLLER PERFORMANCE IN PRESENCE OF MEASUREMENT NOISE In this section, performance of the proposed control strategy in the presence of measurement noises is investigated. The viral load measurements have log-normal noise, therefore such a noise with mean µ = 0.1 and standard deviation σ = 10( −0.21−[0.24×log10 ( i )]) in which i is the number of viruses present in the sample has been considered.

53

Measurements of CD4+ T-Cells are also

subject to a nonlinear type of measurement noise. If the measurements are made by flow cytometry count, it is dominated by a Poissonian process. Therefore the measured count is considered as a Poisson random variable with lambda equal to the nominal concentration. The closed-loop response for the noise corrupted measurements, are shown in Fig. 2. Simulation results show that the proposed control approach is able to control the HIV infection satisfactorily and the outputs approach to the neighborhood of their desired set points (Fig. 2 (a) and (b)). The related drug efficacies are shown in Fig. 2 (c) and (d).

Figure 2


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5.3 CONTROLLER PERFORMANCE IN PRESENCE OF OBSERVER DYNAMICS In this section the results of implementation of the observer based feedback linearizing control strategy are presented. The initial conditions for the observer states are set to:

xˆ ( 0 ) = 0.9 x ( 0 ) cells µL-1 , yˆ ( 0 ) = 0.9 y ( 0 ) cells µL-1 , vˆ ( 0 ) = 0.9 v ( 0 ) copies ml-1 Controllers gains are obtained by maximizing the performance index (37), using the pattern search technique and given below: k 1 = 1642.7 , k 2 = 561.25 , k 3 = 52.66 , k 4 = 8.84 , k 5 = 6.26 The kLNO matrix is determined based on the pole placement technique and is as follows:

k LNO

1.04 0  =  0.04 0   0 1 

The simulation results of the proposed scheme are shown in Fig.3. As can be seen the viral load drops under 50 copies/ml and the target CD4+T cells concentration increases from its initial value to a neighborhood of its normal value in less than 60 days (Fig. 3 (a) and (b)). Thus, the designed control law fulfills the desired objectives. The related drug efficacies are shown in Fig. 3 (c) and (d).

Figure 3 5.3.1 RESULTS FOR CASE OF SPARSE MEASUREMENTS The results presented in previous part were obtained based on the assumption of continuous availability of the outputs which is not true in practice. A blood analysis is necessary for measurements of CD4+ T-cells and the viral load, which can be done periodically for example biweekly or monthly. To take into account this limitation, a continuous discrete time observer has been used.54-58



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For implementation of the continuous-discrete LNO, two steps should be taken: Step 1: The prediction step in the semi-open time interval t ∈ [t k ; t k +1 ) :

xˆ& = f (xˆ ) + g (xˆ )u

(38)

Step 2: The correction step at time t = tk+1:

xˆ (t k +1 ) = xˆ (t k−+1 ) + O −1 (xˆ )k LNO ( y − h (xˆ (t k−+1 )))

(39)

The state estimates are updated in two different ways: i) in the absence of measurement state, estimates are obtained by integrating the equation (38), ii) when measurements become available, state estimates are updated according to equation (39). To see the impact of application of the continuous-discrete observer on the performance of the proposed control scheme, simulations were performed for a typical sampling time of Ts= 15 days.3 In Fig. 4, the closed loop response of the treatment has been shown. From this figure, it can be seen that the performance of the controller is deteriorated but the results are acceptable, because the viral load and CD4+T cells have reached to their desired values within the specified period of time (Fig. 4 (a) and (b)). The related drug efficacies are shown in Fig. 4 (c) and (d).

Figure 4 5.4 RESULTS OF USING DRUG DOSAGES AS MANIPULATED VARIABLES In a real situation for HIV treatment, the manipulated variables are train of impulses (drug doses) because the drugs are taken as pills. The results presented in previous parts were obtained based on the efficacies of the drugs. However, it is difficult to apply the proposed control strategy from practical point of view because drug efficacy is not expressed in terms of drug doses. To cope with this problem, PK and PD models have been included in the HIV model in order to relate drug efficacy to drug dosage. In what follows a practical HIV treatment based on


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the work of Rivadeneira and Moog 38 will be considered. As mentioned in section 4, three widely used drugs RDV, 3TC and ZDV are considered for HIV treatment and the drug consumption of ZDV is fixed to its minimum value because of its high side effects. It is assumed that drugs are taken every 12 hr. In this section, model (1) is modified in order to take into account PK and PD effects. The applied model is as follows: C 2 (t ) C 3 (t )  +  IC 502 IC 503  x& (t ) = λ − dx (t ) − β (1 − ) x (t )v (t ) C 2 (t ) C 3 (t )  + 1+  IC 502 IC 503  C 2 (t ) C 3 (t )  +  IC 502 IC 503 & = β − y ( t ) (1 ) x (t )v (t ) − ay (t ) + λ y  C ( t ) C ( t ) 3 2  + 1+ IC 502 IC 503   C (t ) v& (t ) = γ (1 − 1 1 ) y (t ) − ωv (t ) IC 50 + C 1 (t )  & D1 C 1 (t ) = − k 1 C 1 (t ) C& (t ) = − k D 2C (t ) 1 2  2 D1   ∆C 1 (τ k ) = k 2 D1 t = τ k , k = 0,1, 2, K  D   ∆C 2 (τ k ) = k 2 2 D 2

(40)

where D1 and D2 are inputs of the system that are the drug doses of RDV, 3TC, respectively and τ k − τ k −1 = 0.5 day . As it can be seen, the manipulated variables D1 and D2 have impulsive nature. In model (40), pharmacokinetic models are approximated by two first order differential equations to facilitate the implementation of the proposed control method. Parameters k 1D1 , k 1D2 ,

k 2D1 and k 2D2 have been obtained by minimizing errors between the output of approximated model and the outputs of the exact model 31 and given below:

k 1D1 = 2.4 , k 1D2 = 3.34 , k 2D1 = 1.63 × 10−4 L-1 and k 2D2 = 0.0057 L-1 .



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It should be noted that the above approximations result in 4.5 % and 0.2 % errors in the average efficacies u 1 and u 2 for the standard drug consumptions 150 mg and 600 mg taken every 12 hr, respectively.

Therefore the above approximation can be justified. Furthermore, to

minimize the side effects of ZDV, its consumption is determined in such a way that its average concentration becomes C 3,min = 0.00659 mg L-1 (as given before) over the interval of 12 hrs. The corresponding drug dose is D3,min =13.8 mg which has obtained by using PK model given by Hadjiandreou, et al. 20. Based on model (66), a feedback linearizing controller has been designed. For implementation this control strategy, the system states ( x , y ,v ,C1 ,C 2 ) are required. Since only the system outputs are available, a continuous-discrete LNO with a sampling time of Ts= 15 days is designed for estimation of unmeasured states. The simulation results of the proposed scheme are shown in Fig. 5. As can be seen the viral load drops under 50 copies/ml and the CD4+T cells concentration increases from its initial value to the neighborhood of its desired value (Fig. 5 (a) and (b)). Thus, the designed control law fulfills the desired objectives. The related drug doses are shown in Fig. 5 (c) and (d).

Figure 5 6. CONCLUSIONS In this paper, design of an observer-based control scheme using feedback linearization technique for control of HIV infection is addressed. The observer proposed in this study is a Luenberger-like nonlinear observer. In order to develop an effective drug regimen, three widely used drugs are considered. The concentration of the drug with the highest side effects is set to its minimum value and the controllers parameters are obtained in such a way that the side effects of


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the other two drugs are minimized. The Lyapunov stability technique is used to establish the asymptotical stability of the proposed control scheme in the presence of the observer dynamics and input limitation. The performance of the suggested HIV treatment is illustrated via numerical simulations. It is shown that the proposed control strategy is able to control the HIV infection in the case of model parameter uncertainties, sparse measurements, noise corrupted outputs and impulsive nature of manipulated variables. However for testing the effectiveness of the proposed scheme in practice, it should be applied to real patients for HIV treatment.



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APPENDIX A: CLOSED LOOP STABILITY OF THE PROPOSED CONTROL STRATEGY Consider the following Lyapunov function candidate: V(x% , y% ,v%, e1 , e 2 , e 3 , e 4 , e 5 , θ1 , θ 2 ) = Vo ( x% , y% ,v% ) + V1 (e1 , e 2 , e 3 , θ1 ) + V2 (e 4 , e 5 , θ 2 ) where x% = x − xˆ ,

y% = y − yˆ

and

v% = v −vˆ

are

the

estimation

(41) errors,

1 θ 1 2 ( x% + y% 2 + v% 2 ) , V1 (e1 , e 2 , e 3 ,θ1 ) = (e12 + e 2 2 + e 32 + 1 ) and 2 γ1 2 2

Vo ( x% , y% ,v% ) =

1 2 θ2 2 2 V2 (e 4 , e 5 ,θ2 ) = (e 4 + e 5 + ). 2 γ2 The time derivative of the Lyapunov V function is:

& =V & (x% , y% ,v%) + V & (e , e , e ,θ ) + V & (e ,e ,θ ) V o 1 1 2 3 1 2 4 5 2

(42)

In what follows time derivatives of the individual Lyapunov functions are calculated.

& (x% , y% ,v%) = x% x&% + y% y&% +v%v&% V o

(43)

Using Eqs. (8) and (19) we have:

ˆ ˆ ) − k 1,′ LNO y%1 x&% = −d x% − β (1 − u1 )(xv − xv

(44)

ˆ ˆ ) − a y% − k 2,′ LNO y%1 y&% = β (1 − u1 )(xv − xv

(45)

v&% = (1 − u 2 )γ y% − ω v% − k 3,′ LNO y% 2

(46)

where y%1 = y 1 − yˆ1 and y% 2 = y 2 − yˆ 2 . Noting that y% 1 = x% + y% and y% 2 = v% and using of Eqs. (44)-(46) in equation (43), we get:

& (x% , y% ,v% ) = −(d + k ′ )x% 2 − (a + k ′ ′ %2 %2 V o 1, LNO 2, LNO ) y − (ω + k 3, LNO )v ˆ ˆ )( y% − x% ) + γ y% v% (1 − u 2 ) − ( k 1,′ LNO + k 2,′ LNO ) x% v% + β (1 − u 1 )(xv − xv ˆ ˆ )( y% − x% ) − γ y% v% u = − X% T A X% + β (1 − u )(xv − xv o

1

2


(47)

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 d + k 1,′ LNO 0 k 1,′ LNO + k 2,′ LNO  where Ao =  0 a + k 2,′ LNO γ  0 0 ω + k 3,′ LNO 

  T %  and X = [ x% , y% ,v% ] .  

Using assumptions 2 and 3, and inequalities x% ≤ X% , y% ≤ X% and v% ≤ X% we have:

& ≤ −ψ X% V o

(48)

2

− 2β (1 + u1,max ) γ L − γ u 2,max ) and λAmin where ψ = (λAmin is the minimum eigenvalue of Ao . o o The time derivatives of the V1 and V2 are given by:

& & = e e& + e e& + e e& + θ1θ1 V 1 1 1 2 2 3 3

(49)

& & = e e& + e e& + θ 2θ 2 V 2 4 4 5 5

(50)

γ1

γ2

Using equation (18) we have: e&1 = e 2

(51)

e&2 = e 3

(52)

e&4 = e 5

(53)

Time derivatives of e 3 and e 5 can be obtained as described below. Form Eqs. (18) and (19), we get:

y&&ˆ1 = −d {λ − d xˆ + k 1,′ LNO ( y 1 − yˆ1 )} − a{−a yˆ + λy + k 2,′ LNO ( y 1 − yˆ1 )} + (k 1,′ LNO + k 2,′ LNO ){d (xˆ − x ) + a ( yˆ − y ) − (k 1,′ LNO + k 2,′ LNO )( y 1 − yˆ1 )} ˆ ˆ (d − a )(1 − u1 ) + β xv


(54)


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y&ˆ 2 = (1 − u 2 )γ yˆ − ω vˆ + k 3,′ LNO ( y 2 − yˆ 2 )

Page 28 of 37

(55)

Noting that y&&1,d = y&&ˆ1 − e&3 , u1 ≈ u d ,1 (t ) + θ1 where θ1 is the estimate of θ1* and using Eqs. (26), (28), (54), we have:

e&3 = −k 1e1 − k 2e 2 − k 3e 3 − β (d − a )θ1xˆ vˆ + + (k 1,′ LNO + k 2,′ LNO )(λ + λy ){−d (x − xˆ ) − a ( y − yˆ )}

(56)

− (k 1,′ LNO + k 2,′ LNO )(λ + λy )(d xˆ + a yˆ ) + υ11 Using Eqs. (27), (29) and (55) and the fact that y& 2,d = y& 2 − e&5 and u 2 ≈ u d ,2 (t ) + θ 2 where θ 2 is the estimate of θ 2* , time derivative of e5 can be obtained as given below:

e&5 = −k 4e 4 − k 5e5 − θ2γ yˆ

(57)

& can be written as: Having the time derivatives of e1, e2 and e3, V 1 & = e e + e e + e {−k e − k e − k e − β (d − a )θ xˆ vˆ + V 1 1 2 2 3 3 1 1 2 2 3 3 1 + (k 1,′ LNO + k 2,′ LNO )(λ + λy ){−d (x − xˆ ) − a ( y − yˆ )} − (k 1,′ LNO + k 2,′ LNO )(λ + λy )(d xˆ + a yˆ ) + υ11} +

θ1θ&1 γ1

(58)

= −e123T A1e123 + e 3{−(k 1,′ LNO + k 2,′ LNO )(λ + λy )(d xˆ + a yˆ ) + υ11} {− β (d − a ) e 3θ1 xˆ vˆ +

where e123 = [e1 , e 2 , e 3 ]

T

θ1θ&1 } + e 3{(k 1,′ LNO + k 2,′ LNO )(λ + λy ){−d (x − xˆ ) − a ( y − yˆ )}} γ1

0 − 1 0  0 − 1  and k1, k2 and k3 are positive constants. , A1 = 0  k 1 k 2 k 3 

If we choose υ11 as:

υ11 = (k 1,′ LNO + k 2,′ LNO )(λ + λy )(d xˆ + a yˆ )

(59)

and the following adaptive law for θ1

θ&1 = −k θ′,1 θ1 + e 3 γ 1 β xˆ vˆ (d − a )


(60)

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where k θ′,1 = k θ ,1 γ 1 and k θ ,1 > 0 , equation (58) becomes:

& = −e T A e − k θ 2 + e {(k ′ ′ ˆ ˆ V 1 123 1 123 θ ,1 1 3 1, LNO + k 2, LNO )(λ + λy ){−d (x − x ) − a ( y − y )}}

(61)

Using inequalities e 3 ≤ e123 , x% ≤ X% and y% ≤ X% we have: a a & ≤ −λ min e 2 + e % V 1 A1 123 123 X (d + a )(λ + λ y )(k 1, LNO + k 2, LNO )

(62)

where k 1,aLNO = k 1,′ LNO , k 2,a LNO = k 2,′ LNO and λAmin is the minimum eigenvalue of A1 . 1

& can be obtained as given below: By the same procedure V 2 & & = e e + e (−k e − k e − θ γ yˆ ) + θ 2θ 2 V 2 4 5 5 4 4 5 5 2

γ2

= −e

T 45

θ θ& A 2 e 45 + (−θ 2 γ yˆ e 5 + 2 2 ) γ2

(63)

 0 −1  T where e 45 = [e 4 , e 5 ] , A 2 =   and k4 and k5 are positive constants.  k4 k5 

The following update law is chosen for θ 2

θ&2 = −k θ′,2 θ 2 + e 5 γ 2 γ yˆ

(64)

where k θ′,2 = k θ ,2γ 2 and , k θ ,2 > 0 . Using the inequality e 5 ≤ e 45 and adaptive law (64), we have: (65)

& ≤ −λ min e 2 − k θ 2 V 2 A2 45 θ ,2 2

where λAmin is the minimum eigenvalue of A2 . 2 Based on inequalities (48), (62), and (65) and equation (42), we have:



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& X% , e , e , e , e , e , θ , θ ) ≤ −ψ X% V( 1 2 3 4 5 1 2

2

2

2

− λAmin e123 − λAmin e 45 − k θ ,1θ12 1 2

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(66)

− k θ ,2θ 2 2 + (d + a )(λ + λy )(k 1,aLNO + k 2,a LNO ) X% e123

The above inequality can be rewritten as:

& X% ,e , e ,e , e , e ,θ ,θ ) ≤ −ΛT ΓΛ V( 1 2 3 4 5 1 2

(67)

where

    Γ=    

ψ

− (d + a )(λ + λy )(k 1,aLNO + k 2,a LNO )

0

λAmin

0

0

0

0

0

0

λAmin

0

0

0

0

k 1θ

0

0

0

0

1

2

0   X%     0  e123   0  and Λ = e 45     0   θ1  θ  k 2θ   2  

The observer poles can be assigned such that ψ becomes a positive constant. Since λAmin and 1

λAmin are also positive, it can be concluded that Γ is a positive definite matrix and consequently 2

& X% , e , e , e ,e ,e ,θ ,θ ) < 0 and therefore asymptotical stability of the proposed control V( 1 2 3 4 5 1 2 strategy for the HIV treatment is established. Since e 2 and e 5 converge to zero, we have

yˆ1 → y d ,1 and yˆ 2 → y d ,2 as t → ∞ . Additionally x% , y% , v% also converge to zero resulting in x → xˆ , y → yˆ and v → vˆ as t → ∞ and therefore yˆ1 → y 1 and yˆ 2 → y 2 , leading to convergence of outputs to their desired values.


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APPENDIX B: CONTROLLABILITY AND OBSERVABILITY The system given by equation (7) is controllable if rank of the following matrix is 3. W C (x ) =  g 1 (x ), g 2 (x ), [ g 1 , g 2 ] , ad g21 , g 2  , ad g22 , g 1  , [ f , g 1 ] , [ f , g 2 ] , ad f2 , g 1  , ad f2 , g 2     (68)

The minor whose columns are g 1 (x ), [ g 1 , g 2 ] , [ f , g 1 ] is β 3γ x 3v 3 (d − a) . Therefore rank of the above matrix is 3 if a ≠ d , x > 0 and v > 0 . The first two conditions are satisfied and the last condition means that the viral load should be always positive. Since an infected individual can never get rid of the infection again 39, the last condition is also satisfied indicating that the system is controllable. For the system under consideration given by equation (7), the observability matrix is as follows: dL0f (h1 ) dL0f (h2 )    W O (x ) = dL1f (h1 ) dL1f (h2 )  dL 2 (h ) dL2 (h )  2  f  f 1   1 1 0 0 0 1   = −d −a 0 0 γ −ω   βv (d − a ) + d 2 a 2 β x (d − a ) βγ v − γ (a + ω ) βω x + ω 2   

(69)

The minor formed by second, fifth and sixth columns of W O (x ) is βγ 2 x which is non zero because x > 0 . Therefore it can be concluded that the system is observable.



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AUTHOR INFORMATION Corresponding Author E-mail:[email protected]


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(41) Hajizadeh, I.; Shahrokhi, M. Observer-Based Input-Output Linearization Control of HIV Infection. The 8th International Chemical Engineering Congress & Exhibition (IChEC 2014), Kish, Iran, February 2014. (42) Zarrabi, M. R.; Farahi, M. H.; Efati, S. Using sliding mode control in stability treatment of HIV disease. Adv. Model. Optim. 2012, 14, 165-173. (43) Melgarejo, M. A.; Peña-Reyes, C. A.; Sanchez, E. A genetic-fuzzy system approach to control a model of the HIV infection dynamics. IEEE International Conference on Fuzzy Systems, Vancouver, BC, 2006. (44) Zarei, H.; Kamyad, A. V.; Heydari, A. A. Fuzzy modeling and control of HIV infection. Comput. Math. Method. M. 2012, 2012. (45) Assawinchaichote, W.; Junhom, S. H∞ fuzzy controller design for HIV/AIDS infection system with dual drug dosages via an LMI approach. Int. Energy J 2011, 5, 27-33. (46) Banks, H.; Jang, T.; Kwon, H.-D. Feedback control of HIV antiviral therapy with long measurement time. Int. J. Pure Appl. Math 2011, 66, 461-485. (47) Luo, R.; Piovoso, M. J.; Martinez-Picado, J.; Zurakowski, R. HIV model parameter estimates from interruption trial data including drug efficacy and reservoir dynamics. PLoS One 2012, 7, e40198. (48) Kravaris, C.; Chung, C. B. Nonlinear state feedback synthesis by global input/output linearization. AIChE J. 1987, 33, 592-603. (49) Biswas, P. P.; Ray, S.; Samanta, A. N. Nonlinear control of high purity distillation column under input saturation and parametric uncertainty. J. Process Control 2009, 19, 75-84. (50) Biagiola, S. I.; Figueroa, J. L. A high gain nonlinear observer: application to the control of an unstable nonlinear process. Comput. Chem. Eng. 2004, 28, 1881-1898. (51) Mangsuli, P. R.; Rao, N. J. Nonlinear Luenberger-like observers for nonlinear MIMO systems. Asian J. Control 2008, 10, 495-506. (52) Yoon, S.-S.; Park, J.-K.; Yoon, T.-W. Dynamic anti-windup scheme for feedback linearizable nonlinear control systems with saturating inputs. Automatica 2008, 44, 3176-3180. (53) Luo, R.; Piovoso, M. J.; Zurakowski, R. Modeling Uncertainty in Single-Copy Assays for HIV. J. Clin. Microbiol. 2012, 50, 3381-3382. (54) Téllez-Anguiano, A.; Astorga-Zaragoza, C.; Alcorta-Garcıa, E.; Targui, B.; QuinteroMármol, E.; Adam-Medina, M.; Olivares-Peregrino, V. NONLINEAR CONTINUOUSDISCRETE OBSERVER APPLICATION TO DISTILLATION COLUMNS. Int. J. Innov. Comput. I. 2012, 8, 763-778. (55) Astorga, C.-M.; Othman, N.; Othman, S.; Hammouri, H.; McKenna, T.-F. Nonlinear continuous–discrete observers: application to emulsion polymerization reactors. Control. Eng. Pract. 2002, 10, 3-13. (56) Hammouri, H.; Nadri, M.; Mota, R. Constant gain observer for continuous-discrete time uniformly observable systems. 45th IEEE Conference on Decision and Control, San Diego, CA, December 2006. (57) Andrieu, V.; Nadri, M. Observer design for Lipschitz systems with discrete-time measurements. 49th IEEE Conference on Decision and Control (CDC), Atlanta, France, December 2010. (58) Ahmed-Ali, T.; Postoyan, R.; Lamnabhi-Lagarrigue, F. Continuous–discrete adaptive observers for state affine systems. Automatica 2009, 45, 2986-2990.



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Table 1. Maximum model uncertainties handled by the proposed controller Model mismatch

Maximum model uncertainties (%)

Deviation in β

-11, +100

Deviation in a

-26, +7

Deviation in γ

-7,+28

Deviation in λy

-100,+13800

Deviation in ω

-22,+8


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Figure Captions Figure 1. a) Evolution of the viral load, b) evolution of the CD4+T cells concentration, c) RTI efficacy, d) PI efficacy in case of model parameter uncertainty applied at t=100 day.

Figure 2. a) Evolution of the viral load, b) evolution of the CD4+T cells concentration, c) RTI efficacy, d) PI efficacy for noise corrupted measurements.

Figure 3. a) Evolution of the viral load, b) evolution of the CD4+T cells concentration, c) RTI efficacy, d) PI efficacy using the LNO observer.

Figure 4. a) Evolution of the viral load, b) evolution of the CD4+T cells concentration, c) RTI efficacy d) PI efficacy in case of sparse measurements.

Figure 5. a) Evolution of the viral load, b) evolution of the CD4+T cells concentration, c) RDV dose, d) 3TC dose for impulsive control strategy.


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