A two-loop robust controller for HIV infection models in the presence of parameter uncertainties

doi:10.1016/j.bspc.2015.02.001

Biomedical Signal Processing and Control

Volume 18, April 2015, Pages 245-253

https://doi.org/10.1016/j.bspc.2015.02.001 Get rights and content

Abstract

A two-loop robust nonlinear controller is proposed to deal with uncertain model parameters in HIV infection models, which are described by nonlinear differential equations of three state variables and antiretroviral drugs. The treatment goal is to suppress the concentration of infected CD4+ T cells to a target value using only the measurement of total CD4+ T cell concentration. The outer-loop controller is designed to achieve the treatment goal for the nominal HIV infection model, while a nonlinear disturbance observer (DOB) controller is employed in the inner-loop to compensate for parameter uncertainties. In the nonlinear DOB controller, a disturbance signal, equivalent to the parameter variation in terms of effect on the output, is canceled by its estimate. Numerical simulations verify that the proposed controller achieves robust performance for reducing the concentration of infected CD4+ T cells even in the presence of parameter uncertainties.

Introduction

Since the proposal of several mathematical models by Perelson and Nelson [1] and Nowak and May [2] to represent the interaction of HIV with CD4+ T cells, a variety of research has been actively conducted by means of the control theory to develop drug therapies for HIV-infected patients. A control method based on Jacobian linearization was employed to reduce the viral load [3]. Using the backstepping method, a nonlinear controller was designed for better control performance [4]. However, the backstepping method usually suffers from the problem of “explosion of terms” due to the repetitive differentiation of the virtual input [5]. Treatment scheduling based on model predictive control (MPC) was also developed for the immune system to suppress the virus [6]. An output feedback MPC approach was then presented to handle cases where the measurements of all state variables are not available [7]. In contrast to the two works based on the standard MPC framework [6], [7], an offset-free MPC algorithm was developed to cope with model errors using the state augmentation approach [8]. Although simulation showed that the treatment goal was achieved through the MPC algorithm, the stability is not theoretically guaranteed in the presence of model errors. Feedback linearization has been employed to control the viral load [9], [10], but this approach generally does not work properly in the presence of model errors, as it relies on the exact cancelation of nonlinear terms. A system theoretic approach was presented to treat HIV-infected patients via analysis of bifurcation and stability [11]. Thymic recovery in HIV patients was also studied using an optimal control approach [12].

The parameter values of an HIV-infection model depend on the patient's infection condition [13], [14]. That is, parameter uncertainties (or parameter variations) may exist in the model. In the aforementioned references, excluding [8], however, various treatment algorithms have been proposed without considering any robustifying control term. As a result, they are unlikely to achieve the treatment goal in the presence of large uncertainties although they may work well for small uncertainties. Besides, many of the reported methods [3], [4], [6], [9], [10], [12] require the measurements of all state variables to implement the control law, while such measurements are not feasible in real clinical situations. Therefore, it would be worthwhile to find the drug efficacy for the treatment of HIV infections in the presence of parameter uncertainties without having to measure all state variables.

In this paper, a two-loop robust nonlinear controller is proposed for HIV infection models with parameter uncertainties, even when not all state variables are available for feedback. Many references [7], [8], [13], [14], [15], [16] employed the total concentration of CD4+ T cells and the viral load as measurable state variables. As will be seen later, on the other hand, the proposed controller requires only the measurement of the total concentration of CD4+ T cells. Thus, the treatment goal of this paper is to suppress the concentration of infected CD4+ T cells to the target value using only the total concentration of CD4+ T cells. To achieve the treatment goal in the presence of parameter uncertainties, a nonlinear controller with a two-loop structure is proposed, as shown in Fig. 1. The outer-loop controller is designed such that the treatment goal can be achieved for the nominal HIV infection model. That is, parameter uncertainties are not taken into account in the design of the outer-loop controller. In this regard, the outer-loop controller is also referred to as the nominal controller. The inner-loop controller, on the other hand, is in charge of compensating for parameter uncertainties. The nonlinear disturbance observer (DOB) controller [17] is adopted as the inner-loop controller. In the nonlinear DOB controller, a disturbance signal, equivalent to the parameter variation in terms of effect on the output, is canceled by its estimate. As a result, the shaded block in Fig. 1 becomes approximately equivalent to the nominal HIV infection model. Therefore, even in the presence of parameter uncertainties, the proposed two-loop controller achieves almost similar treatment performance as that of the nominal closed-loop system.

The proposed controller is designed as follows. First, a state feedback controller is designed using the nominal model under the assumption that all state variables are measurable. Second, a nonlinear state observer is designed based on the nominal model to obtain state estimates only from the measurement of total CD+4 T cell concentrations. An output feedback controller is then constructed by combining the state feedback controller and the nonlinear state observer. This output feedback controller plays the role of the outer-loop controller. Finally, the nonlinear DOB controller is designed to make the uncertain closed-loop system behave like the nominal system.

The paper is organized as follows. In Section 2, a mathematical model of HIV infection therapy is presented, and the treatment goal of the paper is stated. Under the assumption that all model parameters are known exactly, the nominal controller is designed in Section 3. In Section 4, the nonlinear DOB controller is designed to cope with uncertain model parameters. To illustrate the performance of the proposed controller, various simulation results are presented in Section 5. Finally, a conclusion is offered in Section 6

Section snippets

Mathematical model of HIV infection therapy

A deterministic model of HIV infection therapy can be expressed by the following differential equations [3], [9], [10], [13], [14], [16]:

$\begin{matrix} {\dot{x}}_{1} & = s - d x_{1} - β (1 - u) x_{1} x_{3} \\ {\dot{x}}_{2} & = β (1 - u) x_{1} x_{3} - μ x_{2} \\ {\dot{x}}_{3} & = k (1 - u_{P}) x_{2} - {cx}_{3} \end{matrix}$ where x₁, x₂ and x₃ represent the concentrations of healthy CD4+ T cells, infected CD4+ T cells and free virus, respectively. Healthy cells are produced at the rate s, and die naturally at the rate d. They are infected at the rate β through interaction with the virus. The infected cells then die at the

Outer-loop controller

In this section, the outer-loop controller is designed to achieve the treatment goals for the nominal HIV model. Since parameter uncertainties are not taken into consideration, various conventional control techniques may be used to design the outer-loop controller. Among them, input–output linearization [19] is employed since it is a popular nonlinear control strategy that yields excellent performance when there is no model uncertainties. To apply the input–output linearization, define the

Inner-loop controller

In this section, an inner-loop controller is designed to compensate for parameter uncertainties. The change of variables (9) transforms system (3) into

$\begin{matrix} {\dot{z}}_{1} & = z_{2} \\ {\dot{z}}_{2} & = {\tilde{b}}_{p} (x) + {\tilde{a}}_{p} (x) u \\ {\dot{z}}_{3} & = - \tilde{c} x_{3} + \tilde{k} x_{2} \end{matrix}$ where $\begin{matrix} {\tilde{a}}_{p} (x) & : = (μ - d) \tilde{β} x_{1} x_{3} \\ {\tilde{b}}_{p} (x) & : = μ^{2} x_{2} + d^{2} x_{1} - ds - {\tilde{a}}_{p} (x) \end{matrix}$ Using (11), we obtain

$\begin{matrix} {\dot{z}}_{1} & = z_{2} \\ {\dot{z}}_{2} & = \tilde{b} (z) + \tilde{a} (z) u \\ {\dot{z}}_{3} & = - \tilde{c} z_{3} + \tilde{k} \frac{z_{2} + d z_{1}}{d - μ} + (\tilde{k} - k \frac{\tilde{c}}{c}) r_{0} \end{matrix}$

where $\tilde{a} (z) = {\tilde{a}}_{p} (T^{- 1} (z))$ and $\tilde{b} (z) = {\tilde{b}}_{p} (T^{- 1} (z))$ . As the inner-loop controller, the nonlinear DOB controller is employed as follows [17]:

$\begin{matrix} \dot{ζ} & = - c ζ + k \frac{q_{2} + d q_{1}}{d - μ} \\ {\dot{q}}_{1} & = q_{2} \\ {\dot{q}}_{2} & = - \frac{a_{0}}{τ^{2}} q_{1} - \frac{a_{1}}{τ} q_{2} + \frac{a_{0}}{τ^{2}} y \\ {\dot{p}}_{1} & = p_{2} \end{matrix}$

Simulation results

In this section, it is demonstrated through simulations that the proposed controller is capable of providing excellent treatment performance in the presence of parameter uncertainties, while the input–output linearization controller fails to achieve the treatment goal. Throughout this section, it is assumed that r₀ = 0.2 in the treatment goal.

Conclusion

A nonlinear robust controller of two-loop structure was proposed to control the HIV infection model when uncertainties exist in the parameters β, k, c. It was theoretically proved that the proposed controller could explicitly handle parameter uncertainties. It was also verified by various numerical simulations that the proposed controller could achieve the treatment goal for any initial conditions in the presence of parameter uncertainties.

Note that only $\tilde{β}, \tilde{k}$ and $\tilde{c}$ are considered as uncertain

Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0010233).

References (23)

R. Zurakowski et al.
A model predictive control based scheduling method for HIV therapy
J. Theor. Biol.
(2006)
M. Barao et al.
Nonlinear control of HIV-1 infection with a singular perturbation model
Biomed. Signal Process. Control
(2007)
M. Mhawej et al.
Control of the HIV infection and drug dosage
Biomed. Signal Process. Control
(2010)
H. Shim et al.
A system theoretic study on a treatment of AIDS patient by achieving long-term non-progressor
Automatica
(2009)
V. Costanza et al.
Optimizing thymic recovery in HIV patients through multidrug therapies
Biomed. Signal Process. Control
(2013)
J. Back et al.
Adding robustness to nominal output feedback controllers for uncertain nonlinear systems: a nonlinear version of disturbance observer
Automatica
(2008)
A. Perelson et al.
Mathematical analysis of HIV-1 dynamics in vivo
SIAM Rev.
(1999)
M. Nowak et al.
Virus Dynamics
(2000)
I. Craig et al.
Can HIV/AIDS be controlled?
IEEE Control Syst. Mag.
(2005)
S. Ge et al.
Nonlinear control of a dynamic model of HIV-1
IEEE Trans. Biomed. Eng.
(2005)

A. Stotsky et al.

The use of sliding modes to simplify the backstepping control method

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Citation Excerpt :
Consider the following positive definite function It must be noted that, even in the presence of parametric uncertainties, the proposed approach is able to give the same results, as presented by (20) in [11]. In this section, simulations results are presented to show the effectiveness of the proposed controller.
This paper presents an adaptive observer-based robust controller design for HIV infection therapy using the Baskakov operators as a Universal approximator. The purpose of the treatment is to reduce, asymptotically, the number of infected CD4+T cells to zero using the assumption of accessibility to the total number of CD4+T cells. The lumped uncertainties including un-modeled dynamics and parameter uncertainties are modeled with the Baskakov operators. All of the aforementioned uncertainties are assumed as a function of time instead a function of several variables. The Baskakov operators’ coefficients are then adaptively adjusted. It must be noted that Baskakov operators’ have been used for function approximation usually in an offline manner in mathematics. However, the contribution of this paper is using Baskakov operators’ in an online manner for an engineering problem. Also, considering the nonlinearities originated from antiretroviral drugs' dosage limitation is another important novelty of this paper which has not been considered in previous related works. Uniformly ultimately boundedness of the observer estimation error and the systems’ states are guaranteed through Lyapunov stability concept. The performance is evaluated analytically to confirm that the observer transient response performs satisfactorily. Simulation results and a comparison with observer/controller based on radial basis functions neural network (RBFNN) verify that the proposed Baskakov operator-based observer/controller achieves reasonable performance for reducing the number of infected CD4+ T cells even in the presence of un-modeled dynamics and parametric uncertainties.
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A two-loop robust controller for HIV infection models in the presence of parameter uncertainties

Abstract

Introduction

Section snippets

Mathematical model of HIV infection therapy

Outer-loop controller

Inner-loop controller

Simulation results

Conclusion

Acknowledgements

J. Theor. Biol.

Biomed. Signal Process. Control

Biomed. Signal Process. Control

Automatica

Biomed. Signal Process. Control

Automatica

Mathematical analysis of HIV-1 dynamics in vivo

SIAM Rev.

Virus Dynamics

Can HIV/AIDS be controlled?

IEEE Control Syst. Mag.

Nonlinear control of a dynamic model of HIV-1

IEEE Trans. Biomed. Eng.

The use of sliding modes to simplify the backstepping control method