Control 004, University of Bath, UK, September 004 ROBUSTNESS COMPARISON OF CONTROL SYSTEMS FOR A NUCLEAR POWER PLANT L Ding*, A Bradshaw, C J Taylor Lancaster University, UK. * l.ding@email.com Fax: 0604 76067 Keywords: Multivariable Control; Parameter Uncertainty; Robustness Simulation; Optimum Control. Abstract The paper discusses the robustness of a decentralised Proportional-Integral (PI) control system, a continuous-time Robust Inverse Dynamic Estimation (RIDE) control system and a multivariable Proportional-Integral-Plus (PIP) control system, which have been each designed to control a linearized Multi-Input and Multi-Output (MIMO) state-space model derived from an industrial non-linear performance plant model. The corresponding simulation trials are carried out and the simulation results are presented and compared in the paper. In this preliminary study, the parameters of the system are varied for each simulation trial, in order to assess the sensitivity of the various control algorithms to model mismatch. Introduction Typical objectives of a control system include satisfactory closed loop transient and steady-state responses, with sufficient robustness to uncertainty. Here, a satisfactory response means that the plant output y follows or tracks the reference input y r, despite the presence of disturbance inputs to the plant and measurement noise. It is also essential that the tracking succeeds even if the dynamics of the plant change sometime during its operation. The process of holding the plant output close to the reference input is generally referred to as regulation. A control system that has good regulation in the presence of disturbance signals is said to have good disturbance rejection, while a control system that has good regulation in the face of variations in the plant parameters, is said to have low sensitivity to these parameters. Finally, a control system that has both good disturbance rejection and low sensitivity is robust. In this paper, a decentralised Proportional-Integral (PI) control system [], a continuous-time Robust Inverse Dynamic Estimation (RIDE) control system [ ] and a Proportional-Integral-Plus (PIP) control system [4 6], have been each designed [7] to control a linearized Multi-Input and Multi-Output (MIMO) state-space model [], which is derived from a high-order industrial non-linear performance plant model for an Advanced as-cooled Reactor (AR) power plant. The robustness of the decentralised PI control system, the continuous-time RIDE control system and the multivariable PIP control system will be discussed and compared in present paper. In the following section, the highorder linearized MIMO state-space model will be transferred to the transfer function model. The parameters of the denominators of these transfer functions will be varied to carry out the corresponding simulation trials and the sensitivity of these control systems to the variations of these parameters is analysed and compared. The most robust control system is presented in this paper. Transfer Function Models Table lists the nine linear transfer function models derived from the input, output state space model of the system [7]. These transfer functions describe the dynamics for each inputoutput pathway, where ij (s) (i,, and j,, ) indicates the ith output to the jth input..405 s).9s + 0.0466s + 0.00004 ( 765 s.70 s 55.76 s +.74 s + 0.07 s + 5.7647.75.s + 0.750s + 0.0006.7799s 0.697 0.99s + 0.904s + 0.0069.0s + 0.079 06s + 0.765s + 0.0047 6.407s + 0.04.7460s + 54s + 0.0004 0.000s + 0.0460 0.57s + 0.97s + 0.000.7999s +.54s + 0.047 ( s) s +.0s + 0.060s + 0.000574 5 5.0005s.757s +.7 s) 4 s +.0s + 0.060s +.44 ( s) 5 ( Table : Transfer Functions models.
Control 004, University of Bath, UK, September 004 The corresponding poles and zeros are listed in Table. In this case, there is one complex-conjugate pole pair with a negative real component, together with 9 negative real poles. In other words, all the poles are located on the left half side of the s-plane and the open-loop system is stable. Since poles further to the left on the s-plane are associated with dynamic modes that decay faster than those associated with poles closer to the imaginary axis, so the transfer function with the pole furthest from the origin, is particularly sensitive to uncertainty, as discussed in Section 4. (s) Zeros: no zeros Poles: -0.009 and 0.0 (s) Zeros: -.0050 and 0.040 Poles: -.4, -0.007 and -0.00699 (s) Zeros: no zeros Poles: -0.00 and 0.009 (s) Zeros: -0.0074 Poles: -0.0077 and.04 (s) Zeros: -0.007 Poles: -0.0060 and.5477 (s) Zeros: -0.00 Poles: -0.005 and 0.505 (s) Zeros: - Poles: -0.009 and.095 (s) Zeros: -0.0 and 7 Poles: -.0000, -0.0500+0.0076j and -0.0500-0.0076j (s) Zeros:.77-6 and 0.04 Poles: -0.0066, -0.077, -.0000 Control Methods Table : Poles and Zeros. For over fifty years, because of its simple controller structure, classical PI control has remained one of the most popular and most widely used control schemes in almost all industries. Its general properties with regard to effectiveness, simplicity and conditions of applicability are well recognised. However, the tuning of PI controllers for MIMO processes is very complex. It is often not sufficient to apply the more familiar univariate methods directly, because interactions between the control loops are not properly allowed for. Even so, decentralised PI control is one of the most common control schemes for interacting MIMO processes in industry []. Robust Inverse Dynamics Estimation (RIDE) control theory has been developed from the Salford Singular Perturbation Method by incorporating the principles of Pseudo-Derivative Feedback (PDF). This gives a type of multivariable PI controller structure that provides the desired de-coupling and closed-loop dynamics. One feature of the RIDE theory is that an inner-loop control term, referred to as the dynamic trim, effectively de-couples the plant dynamics. Such RIDE control systems have been applied to a number of control problems, including a wingless missile [] and a high performance fighter aircraft []. The Proportional-Integral-Plus (PIP) controller can be interpreted as a logical extension of conventional PI/PID algorithms, but with inherent model-based predictive control action [4 5]. Here, multivariable Non-Minimal State Space (NMSS) models are formulated so that full state variable feedback control can be implemented directly from the measured input and output signals of the controlled process, without resorting to the design of a deterministic state reconstructor or a stochastic Kalman filter. 4 Simulation Trials In this section, the denominator parameters of the transfer functions listed in Table, are varied to simulate system uncertainty. The corresponding time responses of the decentralized PI control system, the continuous-time RIDE control system and the multivariable PIP control system are presented. In this manner, the sensitivity of the three approaches to parametric uncertainty is compared. Since the decentralized PI control system and the RIDE control system are both continuous-time designs, while the PIP approach is a discrete-time algorithm, there are two clocks in the Matlab/Simulink simulation [9]. One is for the continuous-time systems with the sample time equal to zero and the other is for the discrete-time system with the sample time equal to 0. seconds. Table shows the non-linear blocks of rate limits for each input variable. These ensure that the individual channels of each controller protect the system against overload. Control Loops Input (UTJ Temp. Control loop) Input ( Control Loop) Input ( Control Loop) Rising Slew Rate Table : Rate Limits. Rate Limits Falling Slew Rate 0.0-0.0 0.07-0.07 0.05-0.05
Control 004, University of Bath, UK, September 004 Step changes to the reference inputs are employed in the simulation trials. These are: a -5 C step change in the Unit- Boiler Transition Joint (UTJ) temperature at 00 seconds simulation time; a + bar step change in the Half-Unit Valve Differential Pressure () at 400 seconds simulation time; and a +5 bar step change in the Turbine Stop Valve Pressure () at 600 seconds simulation time. In the following subsections, three illustrative simulation experiments are considered. 4. Increasing the Smallest Pole by 50% The transfer function with the smallest pole, which means the pole is furthest to the origin, is the most sensitive transfer function to do the corresponding simulation trials. In Table, we can find that the pole.04, of the transfer function (s), is the furthest pole from the origin and has a significant effect on the characteristics of the system. Hence the transfer function (s) is chosen for the initial simulation trials, in order to analyze the sensitivity of the various control systems to parametric uncertainty. In the first case, the smallest pole.04 of the transfer function (s) is increased by 50%, i.e. to.5057. The corresponding denominator matrix is changed from [.0000.090 0.00] to [.0000.5 0.05]. In this case, the time responses of the three output variables are illustrated in Figure. Here, the light grey solid traces represent the three reference inputs. The dark grey solid traces represent the three outputs of the decentralized PI control system. The dashed traces represent the three outputs of the continuous-time RIDE control system and the dash dot traces represent the three outputs of the multivariable PIP control system. The associated input signals are illustrated in Figure with the same legend. In Figure, it is clear that these three control systems are still stable and that the time responses of the three outputs show good set-point tracking. In fact, the time responses of these control systems do not show a significant difference from the corresponding time responses using the original design parameters. However, it is noticeable that in the UTJ temperature control loop, the decentralized PI control system does yield considerable oscillations and overshoot compared to the other two approaches. 4. Decreasing the Value of Both Poles The second illustrative simulation trial involves both poles of the transfer function (s), i.e..04 and 0.0077. These are reduced by a factor of 5 to become.06 and 0.0459 respectively, that means the two poles are moved away from the origin in the negative coordinate axis. The denominator matrix is changed from [.0000.090 0.00] to [.0000.40 9]. The corresponding time responses of the three output variables are shown in Figure, with the input signals in Figure 4. It can be seen that for the multivariable PIP control system (dash dot traces), the time responses in the UTJ temperature control loop and the control loop show un-attenuated sinusoidal oscillations with these new parameter values in the simulation, i.e. the system becomes unstable. For the decentralized PI control system (solid traces), the time response in the UTJ temperature control loop reaches the steady-state value after 000 seconds simulation time. However, it shows a big overshoot and the speed of the time response is very slow. By contrast, the time responses of the continuous-time RIDE control system (dashed traces) reaches the steady-state values with only small oscillations. So we can say by comparing with the decentralized PI control system and the multivariable PIP control system, the continuous-time RIDE control system shows the lowest sensitivity to these parameter variations. 4. Moving Both Poles to the Origin Finally, the same two poles,.04 and 0.0077 of the transfer function (s), are moved to the origin of the complex s-plane. The corresponding denominator matrix is changed from [.0000.090 0.00] to [.0000 0 0]. The time responses of the three outputs are shown in Figure 5, with the inputs in Figure 6. 45 40 45.5 7.5 7 6.5 65 60 55 50 Figure : Output variables with the Pole Increasing 50%.
Control 004, University of Bath, UK, September 004 6 4 0 0.9 0.7 TV Position 5 0.5 0. Figure : Input variables with the Pole Increasing 50%. 40 40 4 400 6 4 65 60 55 50 -.4. TV Position 5 0.5 0. Figure 4: Input variables with the Pole Decreasing by a factor of 5. 445 45 40.5 7.5 7 6 60 5 56 54 5 Figure : Output variables with the Pole Decreasing by a factor of 5. Figure 5: Output variables with the Pole Moving to the Origin.
Control 004, University of Bath, UK, September 004 5 robust control system by comparing to the decentralized PI control system and multivariable PIP control system discussed in present paper. However, clearly a more comprehensive simulation study, perhaps based on Monte Carlo Simulation (MCS) methods, is required before further conclusions can be drawn. 0.9 5 0.75 0.7 TV Position 5 0.5 0. Figure 6 Input variables with the Pole Moving to the Origin. In Figure 5, it can be seen that for the decentralized PI control system (solid traces), the time responses in the UTJ temperature control loop and the control loop are divergent, i.e. they show unstable behaviour. The time responses of the continuous-time RIDE control system (dashed traces) and the multivariable PIP control system (dash dot traces) both show good set-point tracking and reach the steady-state values smoothly, despite the parameter variations. They show low sensitivity to these parameter variations in this case. 5 Conclusions The paper considered three approaches to control system design, namely: continuous-time decentralised Proportional- Integral (PI) control; continuous-time Robust Inverse Dynamic Estimation (RIDE) control; and discrete-time, multivariable, Proportional-Integral-Plus (PIP) control. Each approach has been applied to a linearized -input, -output state-space model of a nuclear power plant, which has been derived from a high-order industrial non-linear performance plant model for an Advanced as-cooled Reactor (AR) power plant [7 ]. The present paper reports a preliminary simulation study into the robustness of the three control approaches. Here, individual parameters of the simulation model are varied, in order to represent model mismatch three examples are given in the paper. From these simulation trials, we can say that the continuous-time RIDE control system provides the lowest sensitivity to the specified parameter variations and the most It should also be pointed out that, while the PI and RIDE approaches were carefully optimised for the present system, the PIP algorithm utilised to date, represents the default case with unity weights. The authors are presently investigating the performance of an improved PIP controller optimised for this particular system. References [] Alvarez-Ramirez J. & Monroy-Loperena R., A PI Control Configuration for a Class of MIMO Processes, Ind. Eng. Chem. Res., 40, pp. 6-99, (00). [] A. Bradshaw & J.M. Counsell, Design of Autopilots for High Performance Missiles, Proc. Instn. Mech. Engrs., Journal of Systems and Control, 06, pp. 75-4, (99). [] E. Muir & A. Bradshaw, Control Law Design for a Thrust Vectoring Fighter Aircraft Using Robust Inverse Dynamics Estimation (RIDE), Proc. Instn. Mech. Engrs., Journal of Aerospace Engineering, (), pp. -4, (996). [4] P.C. Young, M.A. Behzadi, C.L. Wang & A. Chotai, Direct Digital and Adaptive Control by Input-Output State Variable Feedback Pole Assignment, International Journal of Control, 46, pp. 67-, (97). [5] Taylor, C.J., Chotai, A. and Young P.C., (000), State space control system design based on non-minimal statevariable feedback : Further generalisation and unification results, International Journal of Control, 7, 9-45. [6] Taylor, C.J., McCabe, A.P., Young P.C. and Chotai, A., (000), Proportional-Integral-Plus (PIP) control of the ALSTOM gasifier problem, Proceedings of the Institution of Mechanical Engineers, Journal of Systems and Control Engineering, 4, Part I, 469-40 [7] L. Ding, Comparison of Some New Schemes for the Control of the Boiler of a Nuclear Power Plant, Ph.D. Thesis, Lancaster University, (00). [] H.J. Kim, Robustness Analysis and Development of a Boiler Control System for an Advanced as-cooled Reactor Power Station, M.Sc. Thesis, Lancaster University, (997). [9] The MathWorks Incorporated, Simulink, Dynamic System Simulation for MATLAB, (997).