A PID-Sliding Mode Control Design for a Coupled Tank

Transcription

1 International Conference CRATT 0, Raes, Tunisia 0 A PID-Sliing Moe Control Design for a Couple Tank Ahme RHIF, Zohra Karous, Naceur BenHaj Braiek Avance System Laboratory, Polytechnic School of Tunisia (Laboratoire es Systèmes Avancés (L.S.A, Ecole Polytechnique e Tunisie ahme.rhif@issatso.rnu.tn, zohra.karous@enit.rnu.tn, naceur.benhaj@ept.rnu.tn Abstract The sliing moe control (SMC is strongly requeste ue to its robustness against the isturbances. The principle of the sliing moe control is to force the system to converge towar a selecte surface an then to evolve there in spite of the uncertainties an the isturbances. The surface is efine by a set of relations between the state variables of the system. However, the chattering phenomenon remains the main isavantage of this robust control. As solution to this problem, a PID sliing moe approach is expose an teste in this paper. Keywors sliing moe control, iscontinuous control, multimoel fusion, Lyapunov stability, stabilization.. INTRODUCTION The sliing moe approach is classifie in the monitoring with Variable System Structure (VSS [-3]. The sliing moe is strongly requeste seen its facility of establishment, its robustness against the isturbances an moels uncertainties. The principle of the sliing moe control is to force the system to converge towars a selecte surface an then to evolve there in spite of uncertainties an the isturbances. The surface is efine by a set of relations between the state variables of the system. The sliing surface is efine accoring to the control objectives an to the wishe performances in close loop, the synthesis of the iscontinuous control is carrie out in orer to force the trajectories of the system state to reach the sliing surface, an then, to evolve in spite of uncertainties, of parametric variations, The sliing moe exists when commutations take place in a continuous way between two extreme values u min an u max [4, 5]. First of all, in this case, we will be intereste in the problem of the stabilizing control existence. Then we will give an outline for the state space stabilizing partition construction. For the systems controlle with a sliing moe control (SMC, the Lyaponov function is often requeste [6]. This metho is base on the linearization aroun the balance points an on linear systems per pieces. To etermine the fiels of stability, many theories establishe the fact that the system which trajectory is attracte towars a balance point loses energy graually in a monotonous way. Lyapunov generalizes the concept of energy by using a caniate function V(X which epens on the state of the system [7]. This paper is organize in three parts: first we start by presenting the SMC theory an its stabilization conitions in general case. After that, we escribe the hyraulic process: couple tank. Finally, we expose the process moel as well as the esign of a PID-sliing moe control as a solution to the chattering phenomenon.. STUDIED PROCESS: DESCRIPTION AND MODELING In hyraulic plants [8], we nee often to control the level of the liqui in a tank. Usually, the classical regulator implemente by these controls is PID [9-]. This controller is esigne to obtain the best characteristics in terms of steay state error, response spee an stability. The hyraulic system is existing together with two extra-moules which allow the automatic control of the level an the flow with variable coefficients of the analogical PID controller. The circuits an techniques employe on this system present the same functionalities, aims an performances as that use in the professional fiel. Specifications of the hyraulic process: The hyraulic system with couple tank is compose of two transparent tanks mae in Figure. In the horizontal one (Tank, an electrical water pump is use to transfer the liqui, water, from this tank to the vertical one (Tank. On the concerne process there are two transucers: flow an level. The flow transucer is a blae or turbine flow-meter that provies the system output with a pulse voltage which frequency is proportional to the liqui flow. The level transucer uses the pressure exerte by the water in the first tank (tank to generate an elementary eformation on the inbuilt Strain Gages. Then the liqui level can be euce from a signal conitioner proportional to this pressure. The process moelling: The system with couple tank is consiere as a benchmark for the stuy an the analysis of the hyraulic systems control problems. This esigne evice allows us to examine the control level of the liqui in a tank by varying the flow of the pump P using the valve V or by applying a isturbance using the valve V (Figure.

2 International Conference CRATT 0, Raes, Tunisia 0 h = ( q q A h = ( q q3 A ( Finally, the hyraulic system is moelle by the following state representation (3. Figure. The hyraulic system conception Table presents the various parameters of this hyraulic system. Table. System parameters A Section of tank 0.03 m A Section of tank 0.08 m s Variable section of valve m s Variable section of valve m s 3 Variable section of valve m a Discharge coefficient a Discharge coefficient a 3 Discharge coefficient h Tank level 0.6 m h Tank level 0. m u Input Power v k p Pump Gain 7.5 m 3 /s.v k s Transucer Gain 40 v/m g Gravity constante 9.8 m/sec Using table, the relation between the supply voltage u of the pump an the input flow q can be written as (. q = k pu q = sa g( h h q3 = s3a3 gh The vertical tank (tank can be fille from the horizontal tank (tank via the variable channel by opening the valve V place after the pump which changes the incoming flow. A secon valve V (channel inserte below the vertical tank (tank may change the output flow q. When we consier only the case of level control, we will have q =q. By using the flow equilibrium equation we obtain: ( We consier gives (4. ( p ( h = k u sa g h h A h = s a g h h s a gh y = ksh ( ( 3 3 A (3 x x = with x =h an x =h, equation (3 x ( p ( x = k u sa g x x A x = s a g x x s a gx y = ksx ( ( 3 3 A From equation (4, we can represent this nonlinear system by the following form (5. x = f( x + Bu x = f ( x y = Cx (5 with f x s a g x x C k an B k p ( = (, = s = A A s a s a f ( x = g( x x gx 3 3 A A. REVIEW OF SLIDING MODE THEORY. The sliing moe control synthesis We consier a general non linear system represente by (6 []. ẋ = f(x + g(xu + h(t, x (6 where x represents the state variable, f(x an g(x two non linear state functions an h(t,x a function representing the isturbances. This function is limite by a known scalar function Π (7. (4

3 International Conference CRATT 0, Raes, Tunisia 0 h( t, x Π ( t, x (7 To have a sliing moe control, it is necessary to fix a surface s =s(t,x= 0 an to synthesize a commutation law to force the system to reach the trajectory an then to slie in that surface. The control law can be compose of two parts: u = u 0 + u (8 u 0 the nominal control u the iscontinuous control allowing to reject the isturbances. The iscontinuous control u must satisfy the following conition: g( x u = h( t, x (9 The nominal control The nominal control u 0 (x is esigne for systems with no isturbances. This control preserves the system asymptotic. The system woul be written as follow: x = f ( x + g ( x u (0 0 0 where x 0 is the system trajectory ensure by the nominal control u 0. The iscontinuous control The iscontinous control u may have ifferent forms. In this way we can represent it by: u = - M(xsign(s ( with M(x is a linear function. In this way, the system ynamic by sliing moe is represente in (. x = f ( x + g ( x u 0 ( x ( We notice that this representation is equal to the one in the nominal case [3]. Other ways, there are three ifferent sliing moe structures: in the first one, the commutation takes place on the control unit, the secon structure uses commutation on the state feeback an the thir one is a structure by commutation on the control unit with aition of the equivalent control. The high orer sliing moe control consists in computing the erivative of the sliing variable. This metho allows the rejection of the chattering phenomenon while preserving the robustness of the approach. For that, two algorithms can be use: the twisting algorithm an the super twisting algorithm. A comparison between the two algorithms was achieve an shows that the super twisting algorithm is more efficient than the twisting algorithm, espite the approximate results, since it oes not ensure the same robustness to perturbations [4-6]. The aim of the high orer sliing moe control is to force the system trajectories to reach in finite time the sliing set of orer r ρ efine by: 0 S r = x IR n. : s = s =... = s ( r = 0, r IN (3 with ρ >0, s(x,t the sliing function: it is a ifferentiable function with its (r - first time erivatives epening only on the state x(t (that means they contain no iscontinuities.. Stabilizing conitions of the sliing moe control Here the sliing moe control is esigne for the nonlinear system (4. with x = Ax + Bu +ϕ( x, u y = Cx A IR B IR C = c c c > i = n nxn nx,, ( n; i 0,,..., an ϕ ( x, u the nonlinear part : 0 0 ϕ( x, u = ; ϕ( x, u < Mx ϕn We start by consiering a linear sliing surface (7. (5 (6 s=cx =0 (7 To ensure the existence of such moe we esign a switching control u s that makes the system reach the sliing surface an satisfy the stability conition s s < 0. This control may have several structure, here we choose the following one (. = ks = k s sign(s (8 with k>0. u s To stuy the stability of the sliing moe control for the system (5, we consier a quaratic Lyaponov function V = s, then have to prove thatv = ss < 0. The first erivative of the consiere sliing surface (9 is written below: s = CAx + CBu + Cϕ( x, u (0 as we have: Mx < ϕ( x, u < Mx then CMx < Cϕ( x, u < CMx ss < s ( CAx + CBu + CMx Now, we have to prove that s ( CAX + CBu + CMx < 0 Notice that in the converging phase to the sliing surface we have u = u, then equations (5, (7 an (0 give: s ( T T T ( ( T T T T x C CA C CBkC + C CM x < 0 k > C CBC C CA+ C CM Then, to make the system converge to the sliing surface, we have to ensure the equation ( that guarantees s s < 0. ( k ( BC A MI > + (

4 International Conference CRATT 0, Raes, Tunisia 0 To evaluate the control law that makes the system stable in the reaching phase to the esire state, we use the fact that s s < 0. Equation (6 gives: Mx < ϕ( x, u < Mx CMx < Cϕ( x, u < CMx CAx + CBu CMx < s < CAx + CBu + CMx Then In the reaching phase to the esire state, the following equation (5 gives the stabilizing control law of the system. ( ε > u = ( + + ε < ( CB CA CM x if s 0 ( CB CA CM x if s 0 with ε > 0 an I the ientity matrix. Consier s a sliing surface such as: with C=k s an xr the real state. s Cxˆ ( = (3 xˆ = x xr, x the esire state an To carry out the system on the sliing surface s, we select a iscontinuous control which commutates between two ifferent values: u s = ksign (s, with k a positive constant. To ensure the system stability carrie out by this control, we consier the Lyapunov caniate function v = s, v = ss, then we have to prove that v <0. As we have s = m sign( s we get: ss = m s sign( s. To make v <0 we must choose m>0..3 The PID-Sliing moe control To ensure the convergence of the system to the wishe state, a high level switching control is often requeste which generates the chattering phenomenon. In this way, a PIDsliing surface (4 with a saturation function will be propose in this paper to solve this problem [7, 8]. then, s ( x x r s = α ( x xr + α + α3 ( x xr t (4 t ( x x ( x x r r = α + α + α 3 t t t 0 ( x x r where, λ sign( s if s > Ω u = s λ if s Ω Ω (6 with λ an Ω >0, Ω efines the half bounary layer thickness. 3. SIMULATION RESULTS The simulation results are obtaine using the following parameters: commutation constant k= an the parameter of the sling surface µ = 0.5. Knowing that uring the experiment we keep completely V open an V half close which inuce a isturbance on the tank level, we obtaine figures -7. These results enable us to notice that the system controlle by a PID controller has a very great sensitivity to the isturbances create by V. This isturbance is about 0%. So this result shows us the no robust aspect of the PID (Figure. The secon isavantage of the PID appears in the commutation frequency an the level of the control (about 5 which are very high (figure 3. In another han, by applying the sliing moe control, we notice that the system output is much more stable that ensure the system robustness to the isturbances (figure 4. The isturbances are about 3%; these oscillations are known as chattering phenomenon. In aition, the control level an its commutations frequency are much low than for the case of the PID (Figures 5 an 6. The sharp variations appeare on the sliing surface level (Figure 7, are ue to the abrupt change of the set point. In another han, the PID-SMC with the saturation function gives us excellent results in term of stability an precision of the process output an control evolution. In this han, Figure 8 shows the elimination of the chattering phenomenon in comparison with the Figure 5. Other ways, the control evolution (Figure 9 present a small iscontinuous aspect before reaching the steay state in a short time. This lack of sharp commutations gives goo operating conitions for the actuators. The control law chosen for this approach will aopt the saturation function to reuce the iscontinuity on the sliing surface. The control law will be given by: s u = λ sat (5 Ω Figure. System evolution by PID

5 International Conference CRATT 0, Raes, Tunisia 0 Figure 3. System control by PID Figure 7. The sliing surface variation Figure 4. System evolution by sliing moe Figure 8. System evolution by PID-sliing moe Figure 5. Sliing moe control evolution Figure 9. PID-Sliing moe control evolution Figure 6. The equivalent control Figure 0. The sliing surface variation by PID-SMC

6 International Conference CRATT 0, Raes, Tunisia 0 4. CONCLUSION In this work, we eal with the synthesis of a control law by sliing moe approach using a nonlinear sliing surface. In the first time, we presente the class an the properties of this sliing surface aopte. Then, a sliing moe control using the sliing surface evelope together with stability stuies were elaborate. After that, the PID-SMC approach was evelope an compare with the other expose methos. The simulations carrie on the hyraulic system with couple tank, show the effectiveness of this control (PID-SMC in comparison with the other controls simulate such as sliing moe controller an PID controller. [6] S.Kamath, V.I.George an S.Viyasagar, Simulation stuy on close loop control algorithm of type iabetes mellitus patients, IETE Journal of Research, pp 30-35, Vol. 55, 009. [7] A.Rhif, A Sliing Moe Control with a PID sliing surface for Power output Maximizing of a Win Turbine, STM, Journal of Instrumentation an Control, Vol., pp. -6, 0. [8] A.Rhif, Z.Karous, N.Ben Haj Braiek, A Sliing Moe Observer for a Sensorless Position Control Photovoltaic System, Journal of Scientific an Inustrial Research, Vol.7, pp , 0. REFERENCES [] D.B. McDonal, Lyapunov Optimizing Sliing Moe Control for Linear Systems with Boune Disturbance, Applie Mathematical Sciences, Vol., n.9, pp 90-98, USA, 008. [] N. Benhaj Braiek an F. Rotella, Design of observers for nonlinear time variant systems, IEEE Syst. Man an Cybernetics Conference, vol. 4, pp. 9 5, 993 [3] V.I. Utkin, Variable structure systems with sliing moes, IEEE Transactions on Automatic Control, vol., no, pp. -, 977. [4] A.Rhif, Position Control Review for a Photovoltaic System: Dual Axis Sun Tracker, IETE Technical Review, Vol. 8, pp , 0. [5] A.Rhif, A Review Note for Position Control of an Autonomous Unerwater Vehicle, IETE Technical Review, Vol. 8, pp , 0. [6] Z.Li, Q.Shui-sheng, Analysis an Experimental Stuy of Proportional-Integral Sliing Moe Control for DC/DC Converter, Journal of Electronic Science an Technology of China, vol. 3, n., 005. [7] N. Benhaj Braiek, F. Rotella, M. Benrejeb, Algebraic criteria for global stability analysis of nonlinear systems, SAMS Journal, Vol.7, pp. -7, 995. [8] N.Hung, T.D.Viet, J.S.Im, H.K.Kim, S.B.Kim, Motion Control of an Omniirectional Mobile Platform for Trajectory Tracking Using an Integral Sliing Moe Controller, International Journal of Control, Automation, an Systems, vol. 8, no. 6, pp.-3, 00. [9] Hammou Saari, Bernar Caron, Mohame Tajine, On the Design of Discrete Time Repetitive Controllers in Close Loop Configuration, Automatika,vol. 5, pp , 00. [0] M.Hanmanlu, An Al base Governing Technique for Automatic Control of small Hyro Power Plants, IETE Journal of Research, pp. 9-6, Vol. 53, 007. [] Igor Erceg, Gorislav Erceg, Damir Sumina, Development an Implementation of Control Algorithms for Synchronous Generator, Automatika, vol. 5, pp.95-06, 0. [] A.Rhif, Z.Karous, N.Ben Haj Braiek, A high orer sliing moemultimoel control of non linear system simulation on a submarine mobile, Eigth International Multi-Conference on Systems, Signals & Devices, Sousse, Tunisia, March 0. [3] M.Defoort, T.Floquet, A.Kokosy, W.Perruquetti, A novel higher orer sliing moe control scheme, Systems & Control Letters, Vol. 58, pp. 0 08, 009. [4] A.Rhif, Z.Karous, N.Ben Haj Braiek, A High Orer Sliing Moe Observer: Torpeo Guiance Application, Journal of Engineering & Technology, vol., pp.3-8, 0. [5] W.J.Liu, K.K.Shyu, K.C.Hsu, Design of Uncertain Multi-input Systems with State Delay an Input Deazone Nonlinearity via Sliing Moe Control, International Journal of Control, Automation, an Systems, vol. 9, no. 3, pp , 0.