CHAPTER 2 MODELING OF THREE-TANK SYSTEM

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1 7 CHAPTER MODELING OF THREE-TANK SYSTEM. INTRODUCTION Te interacting tree-tank system is a typical example of a nonlinear MIMO system. Heiming and Lunze (999) ave regarded treetank system as a bencmark problem for reconfigurable control and observer based fault diagnosis. Te linear discrete state space models are essential for model based control tecniques. In tis capter, developments of linear discrete state space models for fault-free and faulty systems against actuator failure are presented. Te tree-tank system proposed by Hou et al (005) is used for analysis and syntesis of single objective and multi-objective reconfigurable control systems explained in Capter and Capter 4.. THREE-TANK SYSTEM DESCRIPTION Te scematic diagram of te coupled tree-tank system is sown in Figure.. It is composed of tree identical tanks wit a circular cross section of area S. Te tanks are interconnected by two cylindrical pipes wit a circular cross-section of area S C and outflow coefficients of tank and tank are a z and a z respectively. Te nominal inflows (q and q ) are located at tank and tank respectively. Te inflow rate can be continuously manipulated from 0 to a maximum flow rate of q max to maintain te tank level of max. Te measured variables are te level of tank ( ), tank ( ) and tank ( ). Te nominal outflow pipe as a cross section S C wit an outflow

2 8 coefficient a z and located at tank. Te control objective is to control level of tank and tank by manipulating te inflow rates q and q. Pump Pump q S q S c a z a z a z Tank Tank Tank Figure. Scematic diagram of tree-tank system in Equation (.) Te tree-tank system represented using te mass balance is given d = dt d = dt q - S a sgn ( - ) g( - ) z S S a sgn ( - ) g( - ) - S a sgn ( - ) g( - ) z z d q + S a sgn ( - ) = g( - ) - S a g dt S z z S (.) te Table.. Te pysical parameters of te tree tank system are presented in

3 9 Table. Pysical parameters of te tree-tank system Parameters Values Tank cross-section area S = 0.07 m Pipe cross- section area S C = S = S = S = m Pipe outflow coefficients a z = 0.5, a z = 0.579, a z = 0.7 Maximum level max = 0.68 m Maximum in-flow rate q max =. 0-4 m /s Te tree-tank system equations involve square-root nonlinearities and te flow-rates become proportional to te square root of te tank level. In control engineering, a normal operation of te system may be around an equilibrium point and te signals may be considered as small signals around te equilibrium. However, if te system operates around an equilibrium point and if te signals involved are small signals, ten it is possible to approximate te nonlinear system by a linear system. Suc a linear system is equivalent to te nonlinear system considered witin a limited operating range (Ogata 004). Te linearization procedure is presented in te following section.. DEVELOPMENT OF FAULT-FREE AND FAULTY MODELS Te discrete state space model development tecnique involves te following steps to obtain fault-free and faulty models:. Linearization of nonlinear equations around te operating point using Taylor s series expansion metod.. Discretization of continuous model for fault-free system.

4 0. Modeling of faulty system by introducing loss in control effectiveness in actuator. 4. Linearization and discretization of faulty system... Linear State Space Model for Fault-free System Equation (.) Te linearized state space model in continuous form is given in x(t) = Acx(t) + Bcu(t) y(t) = Ccx(t) (.) were, b b f f f 0 b b b b f f f, A C b b b f f f 0 d d d f, f, f dt dt dt f f 0 q q S f f B C 0 0 q q f f 0 q q S

5 x o x= x = o x o u q -q o u = = u q -q o were [ 0, 0, 0 ] and [q 0, q 0 ] are steady state operating points of level and flow rate respectively. a z S g a z S g a z S g b =, b =, b = S S S Te linearization tecnique is valid in te vicinity of te operating point. Te above nonlinear system is linearized around te following steady state operating points T [ ] = [ ] m o o o and [ q T T -4 o q o ] = [ ] 0 m / s as given below Te continuous state space model for te parameters in Table. is AC , BC 0 0 and CC Most practical systems are continuous-time systems. However, tey use discrete-time controller to obtain optimum performance.

6 Te discrete state space model is obtained by discretizing te continuous state space model wit sampling period T s = s as given in Equation (.) x(k +) = Ax(k) + Bu (k) y(k) = Cx(k) (.) were A , B and C x(k), u(k) and y(k) are te state, input and output vectors of discrete model respectively... Linear State Space Model for Faulty System Te actuator fault is modeled as a bias fault troug te control effectiveness factor wic represents te gain of actuator. If te gain is zero, ten te actuator is 00% effective. During normal operation, te actuator delivers te control signal witout any loss. Terefore, =0. Wen a fault occurs in te actuator, due to eiter partial blockage or aging, te actuator cannot deliver te control signal witout loss. Suc a failure condition can be represented by a reduced control effectiveness factor. Te magnitude of reflects te severity of actuator fault. Te actuator fault wit 80% loss of control effectiveness in actuator (pump) is considered for simulation. Te continuous and discrete faulty models ave te form given in Equations (.4) and (.5)

7 x (t) = A x (t) + Bcf u (t) f cf f f y (t) = C x (t) f cf f (.4) were x f (t) and y f (t) are te state and output vectors of faulty system in continuous form respectively and faulty system state matrix A cf = A c, faulty system input matrix B cf = (-) B c and faulty system output matrix C cf = C c. x (k+) = A x (k)+b u (k) f f f f f y (k) = C x (k) f f f (.5) were x f (k) and y f (k) are te state and output vectors of faulty system in discrete form respectively. A f, B f and C f are te state, input and output matrices of faulty system in discrete form respectively. given below: Model parameters for linearized system in continuous form are as A AC , B 0 0 and C C cf cf cf C given below: Model parameters for linearized system in discrete form are as A A , B and C C f f f

8 4 From te continuous model parameters, it is clear tat 80% loss of control effectiveness factor (actuator fault) on pump as caused canges in te input matrix, by noticing te value of te first element in te first column canges from (B c ) to.7 (B cf ). Te corresponding canges are reflected in all te elements of input matrix B f..4 SUMMARY Te detailed description of te interacting tree-tank system and first principle model are presented. Te discrete state space models for faultfree and faulty systems are derived using Taylor s series linearization tecnique. In te following capter, te conventional control tecnique using te derived discrete model will be discussed.