CHAPTER 6 FAST OUTPUT SAMPLING CONTROL TECHNIQUE

Size: px

Start display at page:

Download "CHAPTER 6 FAST OUTPUT SAMPLING CONTROL TECHNIQUE"

Dwain Shelton
5 years ago
Views:

1 80 CHAPTER 6 FAST OUTPUT SAMPLING CONTROL TECHNIQUE 6.1 GENERAL In this chapter a control strategy for hyperthermia system is developed using fast output sampling feedback control law which is a type of multirate output feedback technique via its reduced order model. In the earlier part of the chapter, a review of fast output sampling control algorithm is presented and in the later part, design of fast output sampling controller for tumour tissue model with blood perfusion variability and tissue property variability is given. The performance of the designed controller is evaluated by simulations. Based on simulation results conclusions are drawn. 6.2 FAST OUTPUT SAMPLING CONTROLLER Fast output sampling is a kind of MROF in which the system output is sampled at a faster rate as compared to the control input (Datatreya Reddy et al 2007).The control signal is held constant during each sampling interval. It was shown by Chammas and Leondes (1979a) that, if a system is controllable and observable, then for almost all output sampling rates, any self conjugate pole configuration can be assigned to the discrete time closed loop system by piecewise constant output feedback, provided the number of gain changes is not less than the system s observability index. As constant gains are used in this technique practical implementation of this controller is easy. Instead of using the state observer the Fast Output Sampling controller can be

2 81 used to realize the effect of state feedback gain by output feedback and then applied to the discretized system Review of Fast Output Sampling Feedback Control Algorithm A Fast Output Sampling Algorithm proposed by (Herbert Werner 1998a, 1998b) is explained in this section. Consider a continuous system, x(t) Ax(t) Bu(t) y(t) Cx(t) (6.1) where x n, u m, y p, A n n, B n m, C p n, A, B,C are constant matrices of appropriate dimensions. Let (,,C) and (,,C) be the system in Equation (6.1) sampled at and secs respectively. given as The system in Equation (6.1) after sampling by secs can be x(k 1) x(k) u(k) y(k) C x(k) (6.2) Assume (,C) is observable and (, ) controllable with observability index. Now the output sampling interval is divided into N subintervals of length = /N such that N. Sampling the system in Equation (6.1) by the sampling interval, the delta sampled system is represented as x(k 1) x(k) u(k) y(kt) Cx(kt) (6.3)

3 82 The output measurement is taken at time instants t=l where l=0, 1, 2. The control signal in the interval k t (k 1) is given as y(k ) y(k) u(t) [L L L... L ] Ly y(k ) N1 k (6.4) where L is the output feedback gain A stabilizing state feedback gain F is designed for the system (,,C), such that the Eigen values of ( F) are placed inside the unit circle and the closed loop system has desirable properties. To show how the Fast Output Sampling controller in Equation (6.4) can be designed to realize the state feedback gain, a fictitious lifted system is constructed. The lifted system having at time t = k, the input u k = u(k ), the state x k = x(k ) and the output y k as x x u k1 k k y Cx Du k1 0 k 0 k (6.5) where T 0 C T T C C C 0 ;D0 N2 T N1 T j C C j0 If a state feedback gain F is designed such that ( F) has no Eigen values at the origin, we can define the fictitious measurement matrix as

4 83 1 C F(F,N) (C0 D0F) ( F), (6.6) which satisfies the fictitious measurement equation y k = C F x k (6.7) For L to realize the effect of F, it must satisfy the equation. L C F = F (6.8) L FC (6.9) 1 F If the initial state is unknown, there will be an error u k u k Fxkin constructing the control signal under state feedback. One can verify that the closed loop dynamics is governed by u F k1 xk. u 0 LD F u k1 0 k (6.10) Thus one can say that the Eigen values of the closed loop system under a Fast Output Sampling control law are those of those of (LD0 F ). F together with The designed controllers are stabilizing with desired closed loop behavior at the output sampling instants but excessive oscillations may be present at inter-sample instants LMI Formulation The designed controller mainly suffers from two problems. First, it is difficult to control the error dynamics which are brought into the closed

5 84 loop system by reconstructing the states. The second one is that large feedback gains tend to render the system very noise sensitive by amplifying the measurement noise. These large feedback gains may also require an excessive control effort which may saturate the actuator. These effects can be reduced by allowing a small deviation from the exact design solution. This can be achieved by posing the design problem in the form of LMI problem. It is possible to enhance the performance by not insisting that LC F =F exactly instead,allowing a slight deviation in design, this in turn leads to smaller gain i.e. we relax the condition that L does not exactly satisfy the above linear equation and include constraint on the gain L. Thus we arrive at the following inequalities L, LD F, LC F. (6.11) F 3 This can be formulated in the framework of Linear Matrix Inequalities as follows 2 1 I T L L I 0, (6.12) I LD F T (LD0 F ) I 0, (6.13) I LC F 2 3 F T (LCF F) I 0. (6.14) Here, the three objectives have been expressed by upper bounds on matrix norms and each should be as small as possible. The value 1 small means low noise sensitivity, 2small means fast decay of estimation error, 3 small means fast output sampling controller with gain L is a good

6 85 approximation of the state feedback gain (Herbert Werner 1998a). The last inequality represents the constraint that L realize an approximation of state feedback gain F. Taking 3 0 gives an exact solution to LC F =F. If suitable limits for 1 and 2 are known, then these bounds can be kept fixed and 3 is minimized under the constraints. This gives a Fast Output Sampling controller which gives an approximation of the given state feedback gain under the constraints given by 1 and 2. It is noted that closed loop stability requires 2 1; i.e. the Eigen values which determine the error dynamics must lie within the unit circle. The LMI in Equation (6.11) are solved by minimizing the linear objective under LMI constraints using the solver mincx ( ) in the LMI control toolbox in MATLAB. 6.3 APPLICATION OF FOS CONTROLLER VIA REDUCED MODEL FOR HYPERTHERMIA SYSTEM The state space model for the higher order system obtained is given in Equation ( ). Those models are obtained by varying the tumour tissue perfusion and normal tissue perfusion. The original system is of 131 x131 orders, and the system is too large for controller design. The output feedback controller design for the original system is done via a reduced model of order 4. Using the model reduction algorithm, the reduced order model for the higher order system is found. Since it is a temperature process tau is chosen to be 12secs. The sampling interval is selected as 12 secs. The number of sub-intervals N is chosen as 10. The open loop responses of all the four systems are shown in Figure 3.4. The systems after model reduction and similarity transformation are given in Equation ( ). The stabilizing state feedback gains are obtained for each reduced order model such that the Eigen values of

7 86 ( F) are placed inside the unit circle and that the closed loop system satisfies the goals of the feedback control system for hyperthermia treatment. The step responses of all the systems with state feedback gain F are observed. Number of gain changes in one sampling interval is fixed as N = 10 and / N 1.2 secs. The higher order continuous time systems are sampled at a rate 1 and delta systems are obtained. In FOS controller, temperature is monitored at every 1.2 sec ( 1.2 secs) where as ultrasound intensity I(0) is updated only for each 12 sec ( 12 secs). Using pole placement, a stabilizing gain matrix F (1x4) via reduced order model is obtained. Using the aggregation technique and the transformation matrix C a explained in Equation (3.6) state feedback gain F 1 (1x131) for the higher order original system is calculated. The output sampling feedback gain matrix L (1x10) for the given system is obtained by solving LC F =F. This FOS feedback gain matrix is of large magnitude and this may amplify the measurement noise and may require large control efforts. So the LMI constraints in Equations ( ) are solved for different values of 1, 2 and 3 to find the FOS feedback gain matrix L 1 of lower magnitude for the original system via the reduced model. The FOS gain matrices for all the four systems are obtained. L is the output feedback gain matrix with magnitude of In practice, these large gains may amplify measurement noise and it is desirable to keep these values low. This can be achieved by solving the LMI optimization problem. After optimization the gain value reduces and is given as L 1

8 87 System-1: L T L N- Perfusion condition L LL = 10 3 [ ] and after LMI optimization the gain is reduced to L 1LL = [ ] System-2: L T H N - Perfusion condition L LH = 10 4 [ ] and after LMI optimization the gain is reduced to L 1LH = [ ] System-3: H T L N - Perfusion condition L HL = 10 3 [ ] and after LMI optimization the gain is reduced to L 1HL = [ ] System-4: H T H N - Perfusion condition L HH = 10 3 [ ] and after LMI optimization the gain is reduced to L 1HH =[ ]

9 88 The reference temperature for the controller was fixed at 41 0 C and the steady state error in the response is eliminated by using an integrator. The designed FOS controller is put in the loop with the simulated plant the closed loop step response i.e. temperature response and variation of the control signal u with time t for all the four systems are observed. Figure 6.1 Temperature responses for system I using FOS controller (Measurement location at normal tissue) (a) Closed loop temperature response and desired temperature trajectory (b) control effort

10 89 Figure 6.2 Temperature responses for system II using FOS controller (Measurement location at normal tissue) (a) Closed loop temperature response and desired temperature trajectory (b)control effort Figure 6.3 Temperature responses for system III using FOS controller (Measurement location at normal tissue) (a) Closed loop temperature response and desired temperature trajectory (b)control effort

11 90 Figure 6.4 Temperature responses for system IV using FOS controller (Measurement location at normal tissue) (a) Closed loop temperature response and desired temperature trajectory (b) control effort From the simulations it is seen that the FOS controller performed consistently well for different blood perfusions cases. From the Figures 6.1(a), 6.2(a), 6.3(a) and 6.4(a) it is seen that the measured temperature in each case reaches steady state approximately at 400 sec and there after it tracks the steady state without any fluctuations or overshoot. This makes the designed system suitable for online hyperthermia system. From the Figures 6.1(b), 6.2(b), 6.3(b) and 6.4(b) the input power needed was large initially and when the temperature reached the equilibrium point the input power compensated the heat conduction to the surrounding tissue and after 400 sec it reached a constant value. The numerical values shows that the LMI optimization procedure adopted reduces the control effort significantly this avoids cavitations in tissues and hardware limitations.

12 91 Table 6.1 Open loop and closed loop error norms with FOS Controller for the four systems (Measurement location at normal tissue) Systems Blood Perfusion (kg/(m 3 s)) Open loop error norm Closed loop error norm with FOS System 1 L T L N w T =0.5, w N = System-2 L T H N w T =0.5, w N = System-3 H T L N w T =10, w N = System-4 H T H N w T =10, w N = Table 6.2 Open loop and closed loop error norms with FOS Controller for the four systems (Measurement location at tumour tissue) Systems Blood Perfusion (kg/(m 3 s)) Open loop error norm Closed loop error norm with FOS System 1 L T L N w T =0.5, w N = System-2 L T H N w T =0.5, w N = System-3 H T L N w T =10, w N = System-4 H T H N w T =10, w N = Table 6.1 and Table 6.2 illustrate the open loop and closed loop error norms with FOS controller for blood perfusion variation in normal tissue location and tumour tissue location respectively. FOS controller outperforms the POF controller in controlling the higher order hyperthermia system via the reduced model of order four. Even though the same simulation parameters in terms of sampling rate and blood perfusion are used for both the controllers the FOS controller gives the minimum closed loop error norm compared to the POF controller. This is because the output temperatures are monitored faster in FOS controller compared to POF controller.

13 EFFECT OF TISSUE PROPERTY VARIABILITY IN HYPERTHERMIA SYSTEM The other source of uncertainty in modeling and control arise from tissue property variation. These values of thermal conductivity, blood perfusion and density vary from patient to patient and in the same patient at different times. To account for these uncertainties these properties are assumed to be deviated ± 3% from the actual value. Nominal blood perfusion and normal thermal conductivity are tabulated in Table 6.3.A system is simulated with these normal tissue properties. Four more systems are simulated with a deviation of ±3% in thermal conductivity and blood perfusion. Table 6.4 represents the percentage of tissue property variation. Table 6.3 Normal Tissue Properties Parameters with units Thermal conductivity W/(m 0 C) Blood perfusion W b (kg/m 3 sec) Muscle Tumour Table 6.4 Percentage Deviation of Tissue Property Thermal conductivity W/(m 0 C) Thermal conductivity W/(m 0 C) Blood perfusion W b (kg/m 3 sec) Blood perfusion W b (kg/m 3 sec) Muscle +3% -3% +3% -3% Tumour +3% -3% +3% -3%

14 Measurement Location at Normal Tissue The fast output sampling feedback controller is designed for both the normal and deviated systems with 12 secs :N = 10 ; 1.2 secs: FOS feedback gain for system with nominal tissue property and measurement location in normal tissue is given as System with nominal tissue property L System with +3% variation in blood perfusion L System with -3% variation in blood perfusion L System with +3% variation in thermal conductivity L System with -3% variation in thermal conductivity 11 L The lower magnitude of feedback gain L 11 assures a reduced control effort together with reduced noise sensitivity and fast decay of observation error. The designed FOS controller is put in loop with the simulated plants and the closed loop response i.e. temperature response and variation of the control signal u with time t for both the normal and deviated systems are observed. The base line temperature is set to T a and the results are reported as the deviation of temperature from the base line of 37 0 C

15 94 Figure 6.5 Response for FOS controller with nominal blood perfusion (W b =2.9kg/m 3 sec) and normal thermal conductivity (a) open loop response (b) closedloop with FOS controller and desired trajecory (c) control effort Figure 6.6 Response for FOS controller with +3% variation in blood perfusion and normal thermal conductivity (a) open loop response (b) closedloop with FOS controller and desired trajecory (c) control effort

16 95 Figure 6.7 Response for FOS controller with -3% variation in blood perfusion and normal thermal conductivity (a) open loop response (b) closedloop with FOS controller and desired trajecory (c) control effort Figure 6.8 Response for FOS controller with +3% deviation in thermal conductivity and nominal blood perfusion (a) open loop response (b) closedloop with FOS controller and desired trajectory (c) control effort

17 96 Figure 6.9 Response for FOS controller with -3% deviation in thermal conductivity and nominal blood perfusion (a) open loop response (b) closedloop with FOS controller and desired trajectory (c) control effort The response of the system with nominal blood perfusion (W b =2.9kg/m 3 sec) and normal thermal conductivity is shown in Figure 6.5. Figure 6.5(a) shows the open loop response of original system, Figure 6.5(b) shows the closedloop response with FOS controller and desired trajectory Figure 6.5(c) shows the control effort needed to achive the desired response. Figures 6.6, 6.7, 6.8 and 6.9 show the response of system with ± 3% deviation in blood perfusion from nominal and ± 3% deviation in thermal conductivity from normal respectively. Table 6.5 shows the error norms for the simulated system with ±3% deviation in blood perfusion. Table 6.6 shows the error norms for the simulated system with ±3% deviation in thermal conductivity.

18 97 Table 6.5 Simulated Error norms for nominal and ±3% deviated blood perfusion (Measurement location at normal tissue) Blood perfusion W b kg/m 3 sec) T r Openloop T s Error norm T r Closedloop T s Error norm Nominal (W b = 2.9 kg/m 3 sec) % % Table 6.6 Simulated response for ±3% deviated thermal conductivity (Measurement location at normal tissue) Thermal Open loop Closed loop Conductivity K W/(m 0 C) T r T s Error norm T r T s Error norm +3% % Measurement Location at Tumour Tissue The fast output sampling feedback controller is designed for both the normal and deviated systems with 12 secs ; N = 10 ; 1.2 secs and measurement location in tumour tissue. Table 6.7 Simulated Error norms for nominal and ±3% deviated blood Blood perfusion ( kg/m 3 sec) perfusion (Measurement location at tumour tissue) T r Open loop T s Error norm T r Closed loop T s Error norm Normal (2.9 kg/m 3 sec) % %

19 98 Table 6.8 Simulated response for ±3% deviated thermal conductivity (Measurement location at normal tissue) Thermal Conductivity T r Open loop T s Error norm T r Closed loop T s Error norm +3% % Table 6.7 shows the error norms for the simulated system with ±3% deviation in blood perfusion. Table 6.8 shows the error norms for the simulated system with ±3% deviation in thermal conductivity. FOS controller designed for the lower order system is capable of controlling the higher order system without the need for state estimation. The control technique used here manages the power levels of the ultrasonic transducer, so that time temperature response of the hyperthermia system tracks the desired trajectory. It is found that the stabilizing controller designed from the reduced order model and applied to the higher order system performance and stability is guaranteed. The fast output sampling controller designed via the reduced order model is effective even if the order of reduction is many folds less than the original system. In steady state, performance of the controller is comparable where as in transient state there exists a deviation from the desired trajectory which makes the system to give a little higher error norm. This is due to uncertainty in blood perfusion. Achieving the desired trajectory leads to treatment precribability and this gives clinical acceptance to hyperthermia. To give clinical acceptance to hyperthermia the error norm should be as small as possible. To reduce the error norm and to give clinical acceptance to hyperthermia treatment DSMCFOS is designed, but the performance of the FOS controller is better compared to the POF counterpart.

CHAPTER 4 STATE FEEDBACK AND OUTPUT FEEDBACK CONTROLLERS

54 CHAPTER 4 STATE FEEDBACK AND OUTPUT FEEDBACK CONTROLLERS 4.1 INTRODUCTION In control theory, a controller is a device which monitors and affects the operational conditions of a given dynamic system.