The penalty matrix Q1 has to be transformed accordingly as Q1 =
Q1 0 0 Q3
. (5.25)
The LQG controller for the system in (5.24) becomes u(t) = −Lx(t)
= −L1x(t)− L2z(t) (5.26)
= −L1x(t)− L2
t−1 s=0
[y(s)− uc(s)].
The size of the penalty matrix Q3 affects the strength of the integrating term in the controller.
Faster servo responses on changes of the reference signal can be obtained by introducing a direct term from uc(t) in (5.26) as
u(t) =−L1x(t)− L2
t−1 s=0
[y(s)− uc(s)] + L2uc(t). (5.27)
the controller design from the very beginning. The controller hence aims at ensuring all performance specifications, while handling the saturation constraints by the actuator. The other possibility is to separate the design of the controller and the design of the antiwindup compensator.
The controller does not take into account the saturation constraints and it is designed to ensure that stability is maintained. It is only when saturation occurs that the antiwindup scheme is turned on. The idea in e.g. [58] is to change the dynamics of the closed-loop system when actuators saturate, so that a good transient behavior is obtained after desaturation, while avoiding limit cycles oscillations and repeated saturations.
In the pole placement design, the windup is overcome by feeding back the actual process input (saturated) instead of the unsaturated (calculated) input signal. If the controller in (5.2) is written in observer form as
Aou(t) = T uc(t)− Syl(t) + (Ao− R)u(t), (5.28) and the saturating actuator is described by the nonlinear function f (.), the controller that avoids windup is then given by [5],
Aov(t) = T uc(t)− Syl(t) + (Ao− R)u(t),
u(t) = f (v(t)). (5.29)
Chapter 6
Included Papers
In this chapter a summary of the papers included in the second part of the thesis is given. Note that the papers have been written to be understood separately and therefore some information is repeated.
Paper I
M. M. Silva, T. Wigren, and T. Mendon¸ca. Nonlinear identification of a minimal neuromuscular blockade model in anesthesia. IEEE Transac-tions on Control Systems Technology, vol. 20, no. 1, pp. 181-188, Jan.
2012.
A new SISO Wiener model with two parameters is proposed for the ef-fect of atracurium in the NMB. An EKF is developed to perform the online identification of the system parameters. This approach outper-forms many conventional identification strategies, and shows good re-sults regarding parameter identification and measured signal tracking, when evaluated on a large patient database. The new method proved to be adequate for the description of the system, even with the poor input signal excitation and the few measured data samples present in this application. The method is of general validity for the identification of drug dynamics in the human body.
Paper II
M. M. Silva, T. Mendon¸ca, and T. Wigren. Online nonlinear identifi-cation of the effect of drugs in anaesthesia using a minimal parameter-ization and BIS measurements. In Proc. American Control Conference (ACC’10), Baltimore, Maryland, pp. 4379-4384, Jun. 30-Jul. 2, 2010.
A new MISO Wiener model with four parameters for the PK/PD of propofol and remifentanil, when jointly administered to patients under-going surgery, is presented. An EKF was used to perform the nonlinear online identification of the system parameters. The results show that both the new model and the identification strategy outperform the cur-rently used tools to infer individual patient response. The proposed DoA identification scheme was evaluated in a real patient database, where the DoA is quantified by the BIS.
Paper III
M. M. Silva. Prediction error identification of minimally parameterized Wiener models in anesthesia. In Proc. 18th IFAC World Congress, Milan, Italy, pp. 5615-5620, Aug. 28-Sep. 2, 2011.
In this paper, PEM algorithms for the identification of Wiener models describing the effect of drugs in patients subject to anesthesia are de-rived. In order to exemplify the performance of the proposed PEM algo-rithms, a database with real records collected from patients undergoing general anesthesia is used. The two parameters of a SISO Wiener model describing the effect of the muscle relaxant atracurium in the NMB are identified. Regarding the DoA, the four parameters of a MISO Wiener model describing the joint effect of the hypnotic propofol and the opioid remifentanil in the BIS are also identified. The results show that the identified parameters give rise to predicted output signals that follow the main trends of the real signals, discarding the noise that highly corrupts the measurements.
Paper IV
M. M. Silva, T. Mendon¸ca, and T. Wigren. Nonlinear adaptive con-trol of the neuromuscular blockade in anesthesia. In Proc. 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC’11), Orlando, Florida, pp. 41-46, Dec. 12-15, 2011.
This paper presents a nonlinear adaptive control strategy based on the Wiener model for control of the NMB in anesthesia. The structure com-bines the inversion of the static nonlinearity present in the Wiener model with a pole-placement controller for the linearized system. The overall strategy exploits identification of a reduced model for the description of the effect of the muscle relaxant atracurium in the NMB. An EKF was developed for that purpose, providing estimates of the model parame-ters for both the linear controller and the blocks where the inversion of the static linearity is performed. Simulations were run in a database of 100 patients simulated with the standard physiologically-based PK/PD model for the NMB. The results show that the nonlinear adaptive con-troller performs well regarding reference following and tackles changes in the patient’s dynamics. Noisy scenarios were also simulated to test the robustness of the proposed strategy.
Paper V
M. M. Silva, T. Wigren, and T. Mendon¸ca. Exactly linearizing adaptive control of propofol and remifentanil using a reduced Wiener model for the depth of anesthesia, to appear in Proc. 51st IEEE Conference on Decision and Control (CDC’12), Maui, Hawaii, Dec. 10-13, 2012.
A closed-loop adaptive controller for propofol and remifentanil admin-istration using BIS measurements is proposed in this paper. The con-troller design relies on a reduced MISO Wiener model for the DoA. The exact linearization of this minimal Wiener structure using the model continuous-time parameter estimates calculated online by an EKF is a key point in the design. A LQG controller is developed for the exactly linearized system. Good results were obtained when the robustness of the proposed controller was assessed with respect to inter and intrapa-tient variability through Monte Carlo simulations on a database of 500 patients.
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Paper I
Nonlinear identi cation of a minimal NeuroMuscular Blockade model in
anesthesia
Margarida M. Silva, Torbjörn Wigren and Teresa Mendonça
Abstract
This paper presents new modeling and identi cation strategies to address the many di culties in the identi cation of anesthe-sia dynamics. The most commonly used models for the eect of muscle relaxants during general anesthesia comprise a high number (greater than eight) of pharmacokinetic and pharmacodynamic pa-rameters. The main issue concerning the NeuroMuscular Blockade (NMB) system identi cation is that, in the clinical practice, the in-put signals (drug dose pro les to be administered to the patients) vary too little to provide a su cient excitation of the system. The limited amount of measurement data also indicates a need for new identi cation strategies. A new SISO Wiener model with two pa-rameters is hence proposed to model the eect of atracurium. An Extended Kalman Filter (EKF) approach is used to perform the online identi cation of the system parameters. This approach out-performs many conventional identi cation strategies, and shows good results regarding parameter identi cation and measured sig-nal tracking, when evaluated on a large patient database. The new method proved to be adequate for the description of the system, even with the poor input signal excitation and the few measured data samples present in this application. It turns out that the method is of general validity for the identi cation of drug dynam-ics in the human body.
1 Introduction
The present paper considers nonlinear identi cation of the dynamics of drugs in the human body. For this purpose, new minimally parameteri-zed models are proposed, overcoming the poor excitation of such prob-lems. Data collected in the surgery room from NeuroMuscular Blockade
(NMB) control cases during general anesthesia is used to exemplify the ideas.
In the clinical practice, the term anesthesia refers to a drug-induced reversible pharmacological state where three main variables must be kept in equilibrium: hypnosis, analgesia and are exia. Hypnosis is de ned as the level of unconsciousness associated with the absence of recall after surgery regarding intraoperative events. Several univariate parameters computed using the raw data from the electroencephalogram (EEG) have been used to monitor the level of hypnosis in patients, namely the spec-tral edge frequency [1], the auditory evoked potentials [2] and the ap-proximate entropy [3]. More recently the Bispectral Index (BIS) [4] has taken the lead, being the index most widely used by anesthetists and researchers in the eld to infer the Depth of Anesthesia (DoA). Anal-gesia is de ned as the absence of pain. However, a quantitative and reliable index for the measurement of pain in patients has not yet been widely accepted and validated. Clinicians use signs such as the presence of tears, changes in heart rate and changes in blood pressure to infer the analgesia condition of the patients. Are exia is de ned as the lack of movement. It is induced and maintained by the administration of muscle relaxants and it aims to achieve an adequate level of paralysis to perform surgical procedures. The NMB can be clinically quanti ed by electrical stimulation of the adductor pollicis muscle in the patient's hand. The blockade level corresponds to the rst single response calibrated by a reference twitch.
In order to address the balance of these three components, the anes-thetists adjust the dose of the corresponding drugs by integrating the NMB and DoA indices with all the other monitored physiological vari-ables. When compared with manual drug administration, automated technologies may carry considerable advantages [5]. If reliable models of the patients' pharmacokinetics (PK) and pharmacodynamics (PD) are available, under or overdosing can be avoided by programming the sy-ringe pumps to target speci c values of the drug eects. Such control softwares need reliable models for the patients' PK/PD as well as au-tomatic identi cation strategies based on those models able to identify the inter- and intra-patients' variability. If set up successfully, these con-trol schemes overcome the drawback of using standardized procedures in drug administration based on population studies.
The PK/PD of the eect of drugs in anesthesia can be modeled as a Wiener model: a linear block in series with a static nonlinearity [6].
The linear part describes the way the drug is diused, accumulated and excreted by the human body. The nonlinear part models the eect of the drug in the patient. Due to this structure, the use of linear models to predict human response to anesthesia is not completely adequate [7].
In [8] a rst approach for reducing the number of model parameters was proposed but a linear model was used to describe the nonlinearity.
Alternatives to the identi cation of linear models can be found in other publications. For example, in [9] a hybrid method based on parameter estimation and an arti cial neural network coupled with a curve tting algorithm was proposed. Good results were obtained but the parameter redundancy is still present in the calculation of the steady-state drug dose prediction. The algorithm proposed in [10] was also previously tested for the anesthesia identi cation case study. The black box model approach was modi ed to take into account the NMB Wiener system speci cations but it still failed in identifying the eight parameters present in the atracurium eect model. Unfortunately, the excitatory pattern of the input cannot be chosen by the user to achieve a better performance of the identi cation methodologies. In general anesthesia procedures, the induction phase usually comprises the administration of a bolus of drug (considered as one nite impulse) and afterwards the measured variables are kept at the desired target values by a low variance drug dose pro le.
Moreover, the available data is limited by the sampling rates accepted by the clinical devices.
From the reasons stated above it is reasonable to assume that, by reducing the number of parameters to describe the system, improved results may be achieved when new system identi cation algorithms are designed. The choice of the appropriate number of parameters should match the parsimony principle [11], that states that the best model to describe a certain system should contain the smallest number of free parameters required to represent the true system adequately.
The main contribution of this paper is hence the use of a minimal number of parameters to model the NMB input-output relation, consis-tent with the excitation present in the available data from real cases.
A nonlinear Wiener model using only two free parameters is proposed for this purpose. From this model a new online adaptive algorithm for parameter identi cation is derived by the use of the Extended Kalman Filter (EKF), providing a second contribution. Experimental evalua-tion of the new algorithm in a previously collected database P of sixty patients undergoing general surgery proves the feasibility of the model
and the algorithm. Due to parameter adaptation, the identi ed signals achieve very good reference signal tracking in test cases, using the afore-mentioned database. A comparison between the identi cation results obtained with the EKF algorithm applied to the minimally parameteri-zed model and the EKF applied to the standard models constitutes the last contribution of the paper. The used metrics show that the tracking errors of both strategies are comparable, hence supporting the choice of the model with the smaller number of parameters as the one that better describes the data, in terms of parsimony. Very close pole zero locations were observed for the standard model, indicating that the available data is not rich enough to excite all modes of that model.
This paper is organized as follows. Section 2 describes the principles concerning the control of anesthesia. Section 3 presents the de nition of both the linear and the nonlinear parts of the new minimally parameteri-zed model. In Section 4 the derivation of the EKF is presented. Section 5 presents the simulation results and a comparison of the new method per-formance with a standard approach for the NMB identi cation, whereas Section 6 gives the conclusions.
2 Control strategies
2.1 Control algorithms
The rst successful attempt to commercialize a device designed to control drug dose administration to patients in a personalized way was 'Diprifu-sor' [12]. The algorithm used by this Target Controlled Infusion (TCI) platform receives information about the patient and then adjusts the drug dose pro le in such a way as to achieve a speci c predicted tar-get eect concentration. The set of PK/PD parameters that are used to compute the drug doses were selected based on population models.
Even with information regarding age, weight and height of the patient, a certain inaccuracy remains, meaning that the anesthetist still needs to readjust the target eect concentration every time the observed drug eect in the patient does not match the clinically desired one. Following this trend several other strategies were recently proposed. For example, in [13] an adaptive approach is proposed to improve TCI based strate-gies. The method combines an optimal variance constrained drug dose design with a hybrid identi cation of the individual patient dynamics.
The model in that study was based on the one described in Section 2.2.
Despite constituting a real improvement regarding the commonly used