Design and implementation of model predictive control for a three-tank hybrid system

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International Journal of Computer cience and Electronics Enineerin (IJCEE) Volume 1, Issue 2 (2013) IN 2320 4028 (Online) Desin and implementation of model predictive control for a three-tank hybrid

1 International Journal of Computer cience and Electronics Enineerin (IJCEE) Volume 1, Issue 2 (2013) IN (Online) Desin and implementation of model predictive control for a three-tank hybrid system Cyril Joseph, Dr. V. I. Geore, and Dr. P. R. Venkateswaran Abstract Hybrid systems are a special class of systems, new enouh to have different definitions dependin on the approach of the researchers. In this paper, systems which exhibit both continuous and discrete dynamics is considered as hybrid systems. Importance of this class of systems nowadays is considerable, iven the fact that diital machines like computers often have to work in conjunction with analo systems. General method of analysis and desin of hybrid systems have not been presented yet, but literature provides many different methods, dependin on the character of the system and aim of control. In this paper a three tank hybrid system is modeled. A model predictive control is desined and implemented for the hybrid model. of the control sinal and difference between the set-point and output value. MPC can be also used for control of hybrid systems; however their diital part must be included. This can be realized by computin the cost function for different confiurations of discrete values and choosin one assurin the smallest result. The alorithm used in this paper is presented in Fi. 1.. Keywords Hybrid system, model predictive control, three-tank system, hybrid modelin. H I. INTRODUCTION ybrid systems are dynamical systems that involve the interaction of different types of dynamics. In this class we are interested in hybrid dynamics that arise out of the interaction of continuous state dynamics and discrete state dynamics. A state variable is called discrete if it takes on a finite (or countable) number of values and continuous if it takes values in Euclidean space Rn for some n >= 1. By their nature, discrete states can chane value only throuh a discrete jump. Continuous states can chane values either throuh a jump, or by flowin in continuous time accordin to a differential equation. Hybrid systems involve both these types of dynamics: discrete jumps and continuous flows. The analysis and desin of hybrid systems is in eneral more difficult than that of purely discrete or purely continuous systems, because the discrete dynamics may affect the continuous evolution and vice versa. The term predictive control refers to a wide rane of control methods which make use of a model of the system to compute the control sinal by minimizin an objective function. The model is necessary to calculate, at a iven instant of time, for which possible future value of the control sinal, the cost function value which would be smallest. Usually, the cost function depends on the chane in the value Cyril Joseph is with the Manipal Institute of Technoloy, Manipal University, Udupi, Karnataka, India. phone: ; ( cyril.cholenkery@mail.com). Dr. V. I. Geore is with the Manipal Institute of Technoloy, Manipal University, Udupi, Karnataka, India.( vi_rect@yahoo.com) Dr. P. R. Venkateswaran is with Weldin Research Institute, BHEL, Thiruchirappalli, , India ( prv_i@yahoo.com). Fi. 1 MPC for Hybrid ystems. II. THE HYBRID THREE TANK YTEM Fi. 2 Three tank system A. Physical system Fi. 2 shows the construction of three tank system the system consists of three cylindrical tanks of equal dimensions. The tanks are connected to each other throuh a manual valve. Three rotameter is fixed at the inlet of each tank to measure the flow rate. Each tank is provided with a Differential Pressure Transmitter (DPT) which measures pressure of the liquid column. This information is converted to 167

International Journal of Computer cience and Electronics Enineerin (IJCEE) Volume 1, Issue 2 (2013) IN 2320 4028 (Online) water level.

2 International Journal of Computer cience and Electronics Enineerin (IJCEE) Volume 1, Issue 2 (2013) IN (Online) water level. To enable various system confiurations, vertical tanks are interconnected by valves, which allows us to work with one, two or three tanks, at a time. The system can be confiured as interactin as well as non-interactin system. B. Hybrid modelin L=0.28m maximum water level in each tank T s =0.01 variable samplin time. III. CONTROL PROBLEM It is required that the level in the third tank be constant h 3 =100mm, while all three output valves v 1, v 2 and v 3 are kept open. Both input pumps are workin. The dynamics of such a three tank confiuration is iven by equations (4), (5) an (6). dh 1 = q 1 V i3 dt 13 2(h 1 h 3 ) V oi 1 2h 1 (4) dh 2 = q 2 V i3 dt 23 2(h 2 h 3 ) V oi 2 2h 2 (5) Fi. 3 Three tank modelin The dynamic behavior of the system can be iven by the followin equations. dh 1 dt = q 1 1 sin(h 1 h 3 )V (h 1 h 3 ) V 1 o1 1 2h 1 (1) dh 2 dt = q 2 2 sin(h 2 h 3 )V (h 2 h 3 ) V 2 o2 2 2h 2 (2) dh 3 = sin(h dt 1 h 3 )V (h 1 h 3 ) + sin(h 2 3 h 3 )V (h 2 h 3 ) V o3 3 2h (3) where: h 1, h 2, h 3 levels in respective tanks 1, 2, 3 cross sections of tanks (tanks dimensions are equal) 13, 23 cross section of diital valves between tanks o1, o2 cross section of output valves q 1, q 2 inflow throuh pumps V 12,V 23 status of diital valves between tanks (0-closed, 1- opened) V 12,V 23 status of output valves (0-closed, 1-opened) ravitational acceleration Given that for the system at hand = 1, 2, 3, i3 = 13 = 23, oi = o1 = o2, i=1,2 and for particular problems, the equations become simpler. They are later on linearized to serve in implementation of the chosen MPC alorithm for the hybrid systems. Basic parameters of the system are: a=0.304m diameter of tanks v=0.015m diameter of diital valves v 3 =0.12m diameter of output valves dh 3 = V i3 dt 13 2(h 1 h 3 ) + V i3 23 2(h 2 h 3 ) V oi 3 2h 3 (6) The equations (4), (5) and (6) are linearized around an operatin point. The state space model of the system is drived as iven equation (7). C 13 C 1 0 C 13 A = 0 C 23 C 2 C 23 C 13 C 23 C 13 C 23 C 3 C = [0 0 1], 1 0 B = D = [0 0], (7) where: C 13 = V i3 13 C 23 = V 23 i3 C 1 = io C 2 = io C 3 = io 2h 10 2h 20 2(h 10 h 30 ) 2(h 20 h 30 ) (8) 2h 30 IV. MODEL PREDICTIVE CONTROL ince MPC makes use of values at specific instants of time, a continuous system with m inputs, q outputs and n 1 states iven by: dx(t)/dt = Ax(t)+Bu(t) (9) 168

3 International Journal of Computer cience and Electronics Enineerin (IJCEE) Volume 1, Issue 2 (2013) IN (Online) y(t) = Cx(t)+Du(t) (10) needs to be discretized: x(k + 1) = Ax(k)+Bu(k) (11) y(k) = Cx(k)+Du(k) (12) CA CA² F= CA 3.. CA Np (17) where x is an input or manipulated variable and y is an output. In order to use the model in the desin of discrete-time predictive control its aumented version should be found. (Matrix D is equal to zero) Takin difference operation at both sides we et: x(k) = Ax(k 1)+Bu(k 1) (13) By definin Δx(k) = x(k) x(k 1) and Δu(k) = u(k) u(k 1) and subtractin from equation (13) we obtain: Δx(k + 1) = AΔx(k) BΔu(k) If we define Δy(k + 1) = y(k + 1) y(k), relatin the output and the state variable is possible because: Δy(k + 1) = CΔx(k + 1)= CAΔx(k)+CBΔu(k) (14) Choosin a new state variable vector x(k) = [Δx(k) T y(k) T ] T we have: Δx(k + 1) A = ot y(k + 1) CA Δx(k) Iqxq y(k) + B Δu(k) CB y(k) = [o Iqxq] Δx(k) y(k) (15) Where Iqxq identity matrix with dimensions qxq o zero matrix with dimensions qxn 1 The prediction of state variable and output variable is calculated as the expected values of the respective variables, hence, the noise effect to the predicted values bein zero. For notational simplicity, the expectation operator is omitted without confusion. Effectively, we have Y = Fx(ki) + φδu (16) Where CB CAB CB 0 0 φ = CA 2 B CAB CB 0... CA Np 1 B CA Np 2 B CA Np 3 B CA Np Nc B (18) The incremental optimal control within one optimization window is iven by: ΔU=(φ T φ + Ṝ) 1 (φ T Ṝs r(ki) φ T F x(ki)) (19) where matrix φ T φ has dimension mnc mnc and φ T F has dimension mnc n, and φ T Ṝs equals the last q columns of φ T F. The weiht matrix Ṝ is a block matrix with m blocks and has its dimension equal to the dimension of φ T φ. The set-point sinal is r(ki) = [r1(ki) r2(ki)... rq(ki)] T as the q set-point sinals to the multi-output system. Applyin the recedin horizon control principle, the first m elements in ΔU are taken to form the incremental optimal control: Δu(ki) = [ Im Om Om] (φ T φ + Ṝ) 1 (φ T Ṝs r(ki) φ T F x(ki)) = Ky r(ki) Kmpc x(ki) (20) where Im and Om are, respectively, the identity and zero matrix with dimension m m. The control sinal is Δu(ki) and the matrices (A,B,C,D) come from the aumented model used for the predictive control desin. With the information of xref (ki) replacin x(ki), the predictive control law is then modified to find ΔU by minimizin; J = Rs Fxref(ki) T Ṝsr (ki) Fxref (ki) 2ΔU T φ T (Rs Fxref(ki) + ΔU T (φ T φ + Ṝ)ΔU (21) The cost function and control sinal are iven by J = Rs Fxref(ki) T Ṝsr (ki) Fxref (ki) 2ΔU T φ T (Rs Fxref(ki) + ΔU T (φ T φ + Ṝ)ΔU ΔU = (φ T φ + Ṝ) 1 (φ T Ṝsr(ki) φ T Fx(ki)) (22) (23) 169

4 International Journal of Computer cience and Electronics Enineerin (IJCEE) Volume 1, Issue 2 (2013) IN (Online) V. IMULATION REULT A. Controller output: Fi. 4 controller output Fi. 4 shows the controller sinal which is iven to the two input pumps. The controller output dies down to zero after the process variable is reached the set point. B. Process variables Fi. 5 shows the level of water in each tank. As mentioned in the control problem, the level of tank 3 is maintained at 100mm. The setpoint is achieved in 2 samplin instants. Fi. 5 level of water in tank 1, tank 2 and tank

5 International Journal of Computer cience and Electronics Enineerin (IJCEE) Volume 1, Issue 2 (2013) IN (Online) REFERENCE [1] Linpin Wan, Model Predictive Control ystem Desin and Implementation Usin MATLAB, priner, London 2009 [2] J.L. Villa, M. Duque, A. Gauthier, N. Rakoto-Ravalontsalama, MLD Control of Hybrid ystems: Application to the Three-Tank Benchmark Problem, 2003 [3] M. Lazar, W. P. M. H. Heemels,. Weiland, A. Bemporad, tabilizin Model Predictive Control of Hybrid ystems, IEEE Transactions on automatic control, Vol. 51, No 11, November 2006 [4] A. Bemporad, W.P.M.H. Heemels, and B. De chutter, On hybrid systems and closed loop MPC systems, IEEE Transactions on Automatic Control, vol. 47, no. 5, pp , May 2002 [5] E. F. Camacho, D. R. Ramirez, D. Limon, D. Munoz de la Pena, T. Alamo, Model predictive control techniques for hybrid system, Annual Reviews in Control , p [6] Eduardo F. Camacho, Carlos Bordons, Model Predictive Control, priner, London

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