Design Linear Quadratic Gaussian for Controlling the Blood Pressure of Patient Noor Safaa Abdul-Jaleel Department of Electrical Engineering Assist. Lect. In Al-Mustansyria University Baghdad, Iraq Abstract: Controlling of mean arterial pressure is investigated in this paper by using Linear Quadratic Gaussian (LQG) for controlling the blood pressure especially for the patient suffering from the sudden change in blood pressure, patients in intensive care or while surgery. Two types of LQG controller are used in this paper; where the two controllers give a good performance for controlling the blood pressure, but LQG/LTR give the system more stability with faster response. Keywords: LQG, LQG/LTR, Blood pressure. 1. Introduction One of the most important roles in the medical implementation is the blood pressure. Where, arterial pressure can be consider as one of the most parameters that play a key role to assist in maintaining the safety of the patient during surgery procedures [1]. That's what refers it from the literature in [2, 3, and 4]. When the surgical is compelled, the patient is transferred to the intensive care unit for the purpose of monitoring what could be exposed to him the patient after the operation from complications, especially developments in the blood pressure changes to ensure control it in a timely manner. It must also injection the medication to the patient carefully by the physician in charge of the anesthesia process control. Control of sudden blood pressure changes requires a controlling unit effective to get a good result, avoiding random noise and reading the wrong measurements [2]. Some literature suggested controllers included adaptive control, optimal control and PI controller. D. A. Linkenes [6] gives simple applications of anesthesia which is used a fuzzy logic control with popularize predictive control to repose the muscles. Also, to reduce or minimize the complications that can be occur after surgery especially with heart patients can use the drug, such as constant injection drug sodium nitroprusside (SNP), which has the ability to decrease blood pressure of the patient quickly, but may show his serious side effects when giving him an overdose [5]. This paper organized as follows: section 2, present the model patient. Section 3, describe the controller design. Section 4, represent the simulation results. Finally, section 5 describes the conclusion..
2. The model of patient: The patient blood pressure model used here was developed as follow [6]:. (1). (2). (3). (4). (5) Where: " m(t) and w(t) are zero-mean the Gaussian random process (noise)." The state space model is: A=. (6) 33.3 4 6 B=, B m =. (7) C = [ 1 ], D = []. (8) 3. The design of controller: 3.1 Linear Quadratic Gaussian (LQG): Linear Quadratic Gaussian (LQG) design problem is stable in optimal random control theory and that has many applications in the modern world which ranges from medical processes controllers and even atomic power plants, ect. [7]. The LQG control system design which is based on the use of a randomly linear quadratic regulator in cascade with a Kalman filter object at minimizing a square cost function include selected states and inputs of the system. The original LQG formulation is such that it functions as a regulator [8].
LQG Control = Optimal observer + Optimal state feedback 1. The optimal state estimator and optimal control designs are solved separately, based on the separation principle. 2. Design the Kalman filter first, and then viewing the estimated states as if they were the actual states, design the optimal state feedback control (LQR). ( ). (9) ). (1) Where is the controller gains: = 1. (11) Where P c is a symmetric positive semi-definite matrix and is the solution of the following Riccati equation [9]: A T p c + P c A - P c BR -1 B T P c + M T QM =. (12) LQG r + y - Plant K Kalman Filter Fig. (1): The block diagram of controller
u + + y K 1/s K + - A K f Fig. (2): Kalman- filter block diagram 3.2 LQG Control with Loop Transfer Recovery LQG controllers, unlike LQR, may have very small stability margins: 1- Small disturbances may drive the system unstable 2- Open-loop transfer function with LQR: ( )=( ) 1.. (12) 3- Open-loop transfer function with LQG: ( )= ( + + ) 1 ( ) 1. (13) 4- Loop transfer recovery technique can be used to reduce the difference between these open-loop transfer function. LTR designs at the system s input and output: 1. The LTR design can be performed at the input of the plant model, by replacing with = + letting. (14) The open-loop transfer function with LQG control approaches the open-loop transfer function with LQR control ( + + ) 1 ( ) 1 = ( ) 1. (15) 2. The LTR design can be performed at the output of the plant model, by replacing with = + letting.. (16)
Phase (deg) Magnitude (db) Output Response SAUSSUREA (ISSN: 373-2525), 216 The open-loop transfer function with LQG control approaches the open-loop transfer function of the Kalman filter ( + + ) 1 ( ) 1 = ( ) 1. (17) 4. The simulation results: The simulation results of the developed controllers to control the blood pressure of the patient are presented in this section. 1.4 Step Response 1.2 1.8.6.4.2 1 2 3 4 5 6 7 8 9 1 Time (sec) Fig. (3): Step Response with LQG controller 5 Bode Diagram From: In(1) To: u1-5 -1 18 9-9 -18-27 1-1 1 1 1 1 2 1 3 1 4 Frequency (rad/sec) Fig. (4): Bode plot of the closed loop system
Output Response SAUSSUREA (ISSN: 373-2525), 216 Step Response From: In(1) To: u1 1.2 1.8.6.4.2 1 2 3 4 5 6 7 8 9 1 Time (sec) Fig. (5): Step Response with LQG/LTR controller The analyzed system described by equations of the controllers. The simulation results show the effect of system with two types of controller. The system with LQG/LTR sow more stabilizes with faster response than with LQG controller. 5. Conclusion: The paper design two types of controllers (LQG and LQG/LTR) for tracking control of the blood pressures for patient in suddenly change of the blood pressure or the patient in surgery case. The performance of each of the controllers was simulated by using matlab/simulink to achieve robust stability. Two controllers give a good performance for controlling the blood pressure, but LQG/LTR gives the system more stability with faster response. References: [1] Taghreed M. Mohammed Ridha, "Model Predictive Control of Blood Pressure by Drug Infusion", IJCCCE Journal, Vol. 11, No. 1, pp. 32-45, 211. [2] R. C. Dorf and R. H. Bishop., "Modern Control Systems", Pearson Education Inc., India, 28. [3] J. B. Slate and L. C. Sheppard, "Automatic control of blood pressure by drug infusion", IEEE PROC, Vol. 129, Pt. A, No. 9, pp. 639-645, December 1982.
[4] R. Meier, J. Nieuwland, A.M. Zbinden and S.S. Hacisalihzade, "Fuzzy Logic Control of Blood Pressure during Anesthesia", IEEE Conference on Decision and Control, Arizona, USA, pp. 12-17, December 1992. [5] Anderson L. Cavalcanti and Andre L. Maitelli, "Design of an Intelligent Adaptive Drug Delivery System for Arterial Pressure Control", WSEAS Transactions on Systems and Control, E-ISSN: 2224-2856, Vol. 1, pp. 74-712, 215. [6] D.A. Linkens, "Adaptive and Intelligent Control in Anesthesia", IEEE Conference on Decision and Control, Arizona, USA, pp. 6-11, December 1992. [6] C. W. Frei, M. derighetti and A. M. Zbinden, " Modelling for control of Mean Arterial Blood Pressure (MAP) During Anesthesia", presented at 2n MATHMOD, Vienna, 5-7 February, 1997. [7] Abdulrahman Kalbat, "Linear Quadratic Gaussian (LQG) Control of Wind Turbines", IEEE, May 31, 213. [8] David J. Smith, and Hemanth V. Porumamilla, "LQG Based Robust Tracking Control of Blood Gases during Extracorporeal Membrane Oxygenation", American Control Conference on O'Farrell Street, San Francisco, CA, USA, June 29-July 1, 211. [9] Jafar Z. A. Montazeri, Mohmmad R. J. Motlagh and Javad Poshtan, "Design and comparison of LQG/LTR and H controllers for a VSTOL flight control system", Journal of the Franklin Institute 344 (27) 577 594, 14 February 26.