Control of a radiative furnace under precise and uncertainty conditions

Transcription

1 Control of a raiative furnace uner precise an uncertainty conitions Electrical Department Islamic Aza University -Zanjan Branch No. 8 - Golestan 7 - Shahrak Ansarieh Zanjan IRAN pay_naz@yahoo.com Abstract: - This paper iscusses the control of a raiative furnace uner precise an imprecise moeling. Regaring nonlinearity nature of the process, in the precise case, feeback linearization technique is applie to the furnace an uner uncertainty conitions, sliing moe control is use. The conitions are completely practical an have gaine from an inustrial process. Key-Wors: Feeback Linearization Raiation Furnace Uncertainty Sliing Moe Control. Introuction Raiative furnace is the useful equipment at inustry. It works with raiation like many other evices [], [], [] an [], of course by heat raiation. It is assume that we encounter with two essential cases. In the first case, we have precise moel an we woul control it by feeback linearization technique an in the secon one, sliing moe control is applie ue to uncertainty of the moel. Now we explain the basic concepts of two aforementione control methos an then simulate them. Feeback linearization is an approach to nonlinear control esign which has attracte a great eal of research interest in recent years. Feeback linearization has been use successfully to aress some practical control problems. Design of a nonlinear control system for a Variable Air Volume Air Conitioning (VAVAC) plant through feeback linearization is presente in [5]. For an inuction motor the sliing moe controller is combine with an aaptive input output feeback linearization technique in orer to preserve the system robustness with respect to stator an rotor resistances variations an uncertainties [6]. In [7] a ecentralize nonlinear controller for large-scale power systems is investigate. The propose controller esign is base on the input output feeback linearization methoology. In reference [8] the technique of feeback linearization is use to esign controllers for isplacement, velocity an ifferential pressure control of a rotational hyraulic rive. Application of a feeback linearization technique for control of a istribute solar collector is escribe in [9]. Aaptive feeback linearization control technique for chaos suppression in a chaotic system is use in []. Disturbance ecoupling an trajectory tracking of nonlinear control systems using an observer-base fuzzy feeback linearization control (FLC) is evelope in []. Application of the feeback linearization to the moel of a power system is investigate in []. A scheme for temperature control of a greenhouse using the feeback linearization is presente in [].. Feeback linearization review Consier a system escribe by x = f ( x) + ug( x) () where f an g are smooth vector fiels on some " open set X R containing an f() =. There exists smooth functions qs, S( X) with s ( x) for all x in some neighborhoo of the origin, an a local iffeomorphism T on R n with T() =. We efine v = q( x) + s( x) () z = T ( x) () The resulting variables z an v satisfy a linear ifferential equation of the form z = Az + bv () ISSN: Issue 5, Volume, May 9

2 B where the pair (A, b) is controllable. In this case, the system is calle feeback linearizable. qx ( ) u = s( x) + s( x) v (5) By applying a state transformation z = M z such that the resulting system is in controllable canonical form z = M AMz + M bv where. M AM = a a a.. an (6) (7) ( ) () z = M T x ' = = ( ) ( ) + ( ) () v v az q x am T x s x u where ' a [ a a... a n ] = (5). Raiative furnace ynamics Fig. shows a typical horizontal raiative furnace which is use for tempere glass prouction. Assuming the temperature in the whole of the glass an the upper an lower walls is uniformly change an the conuctive heat transfer is negligible. M b = : (8) An the a i 's are the coefficients of the characteristic polynomial Figure. Horizontal raiative furnace From raiation heat transfer we have n n si A = s + a s (9) i = A further state feeback form j j T p = a ( Tw Tp ) T w = a[ Tw Fw ptp ( Fw p) TB ] + αp T B = a( Tw TB ) (6) v = v + [ a a... a ] z () n results in the close loop system z = Az + bv () where where T P, T w an T B are glass plate, upper wall an lower wall temperatures respectively. P is the input electric power an a, a an a are as below a = βσfw pa a = ασa a = σαa( F ) (7) w p A.. =, b = : () where the σ, the Stefan-Boltzmann constant is * -8 W/m.K, A is the area of the wall, F w-p is the shape factor between upper wall an plate an α, β are the warming rate of the walls an plate respectively. In simulation following practical ata have been use. ISSN: Issue 5, Volume, May 9

3 F w-p =.7 α = 8* -5 β = 5* -5 A = (8) Now apply feeback linearization to furnace system. We efine state variables as follows T p = x T w = x T B = x (9) Then arrange equations in companion form ( ) ( ) x = f x + g x u () y () = v x () = v (7) u = m( x ) + n( x ) v (8) ( ) m x ( ) n x = Lf h LL h (9) g f = LL h () g f From MATLAB symbolic toolbox we obtain m( x) = βσf x ( x x ) + w p αβσ Fw px x Fw px Fw p x [ ( ) ] () h( x ) = y = x () n(x) = σαβf w-p x () f ( x ) a( x x) a[ x Fw px ( Fw p x ) ] = a( x x ) () For tracking control task efine esire path x an tracking error is e = x - x () an g( x) = α g ( ) ( ) Lhx = g f () LLhx () Thus we have relative egree an since the system orer is, the orer of the internal ynamics is. Now to investigate stability of zero ynamics, efine μ = h(x) = x μ = L f h(x) φ(x) = [ μ μ ψ ] () φ(x) is iffeomorphism so we have L g ψ(x) = ψ(x) = x (5) v = x () - k e () k e () meanwhile, when F w p is one, the parameter a is set to zero an since the relative egree remains an system orer is ecrease to, the I.O.L. converts to I.S.L. an there is no internal ynamics appears in this case. This case occurs when the plate an the wall below have the same size an therefore it is a feasible subject. We use this situation for simplifying the necessary equations uner uncertainty conition an applying sliing moe control as below. In section 5 we woul explain more the sliing moe control in etails. x ) = f ( + f (x) ( x) b( x)u. = x )x ] a x ( x + x) a a [ + ( F x a a w p μ = μ = ψ (x) = a ψ (x) (6) a is a positive value, so zero ynamics is stable an thus we coul apply the input-output linearization. b(x) = ( x + x) a α a (5) ISSN: Issue 5, Volume, May 9

4 . Simulation results with feeback linearization Suppose electrical system of the furnace power coul vary from zero to 6kW linearly. Also initial conitions for plate an walls are o K an 5 o K respectively. Set point changes from 7 o K to 9 o K. 9 Figure. Internal ynamics behavior with two poles at s = -. an step input U(Electric Power) Glass Temperature Figure 5. Electric power behavior with two poles at s = -. an step input Figure. Step response of plate temperature with two pole at s = Glass Temperature Up Temperature Figure 6. Ramp response of plate temperature with two pole at s = Figure. Step response of upper wall temperature with two poles at s = -. Down Temperature Up Temperature Figure 7. Ramp response of upper wall temperature with two poles at s = ISSN: Issue 5, Volume, May 9

5 Down Temperature Now we change the one pair poles of the controller form s =. to s =. an investigate responses. We expect the response be slow that the following simulations isplay this Figure 8. Internal ynamics behavior with two poles at s = -. an ramp input Glass Temperature Figure. Step response of plate temperature with two pole at s = -. U(Electric Power) Figure 9. Electric power behavior with two poles at s = -. an ramp input Now we simulate ramp input response of the furnace system with a step isturbance. U(Electric Power) Figure. Electric power behavior with two poles at s = -. an step input Glass Temperature Sliing Moe Control review Consier the single input ynamic system as the following ynamics ( ) ( ) = +. (6) ( n ) x f X b X u Figure. Ramp response of plate temperature with two pole at s = -. an step isturbance Where the scalar x is the output of interest, the scalar u is the control input an X is the state vector as below ISSN: Issue 5, Volume, May 9

6 X x x = ( n ) x x x = ( n ) x (7) In equation (6) the function f(x) (in general nonlinear) is not exactly known, but the extent of the imprecision on f(x) is upper boune by a known continuous function of X ; similarly, the control gain b(x) is not exactly known, but is of known sign an is boune by known, continuous function of X. The control problem is to get the state X to track a specific -varying state X in the presence of moel imprecision on f(x) an b(x) as the following efinition X (8) For the tracking task to be achievable using a finite control u, the initial esire state X () must be such that X () = X() (9) Let x be the tracking error in the variable x as below x = x x () An let X X X x x x = = ( n ) () Be the tracking error vector. Furthermore, let us efine a -varying surface S(t) in the state-space R (n) by the scalar equation sx ( ; t ) =, where s( X ; t) ( ) n t = + λ () x An λ is a strictly positive constant, whose choice we shall interpret later. For instance, if n =, s = x +λx () Satisfying conition or sliing conition for remaining system trajectories on the surface is s t η s () Where η is a strictly positive constant. Essentially, equation () states that the square "istance" to the surface, as measure by s, ecreases along all system trajectories. 6. Simulation results with sliing moe control Uncertainty in variable α causes an uncertainty in f(x) an g(x). The practical range for α as the following α 9 (5) x~ X ~ = X X = x~ (6) S(x,t) = ( + λ)x ~ (7) t S = x~ + λx~ = û = fˆ + x λx~ (8) ˆ ( x ) f = a + x a [ x + ( F ) x ] a x x a w p (9) fˆ f F (5) F = a x + x a 5 6 ( ) x + ( F ) x a w p (5) For security of the sliing conition, we a a iscontinuous aroun surface s= as below ISSN: Issue 5, Volume, May 9

7 [ û k sgn() s ] u = bˆ (5) Where bˆ = k 5 5 a ( x + x) a ( F ) ( ) ˆ (5) β + η + β u (5) Sliing conition check x Sliing Control for vertical furnace b β = max =.5 b (55) min To choose an appropriate value for k, one may be use a constant value of k that satisfies equation (5) at all s. In case of absence of this fulfillment, it shoul be select a greater k for satisfaction. Also to ecrease input chattering with high s frequency switches, one can uses sat ϕ instea of sgn( s ) in equation (5). sat is saturation element an sgn means sign function or element in MATLAB. In aition, if x, then reach to the x ( ) ( ) sliing surface is relate to the following inequality t reach ( = ) s t (56) η In reach, state path reaches to the sliing surface an then tens to x with constant n that for our problem with n = is. λ λ Now regaring the above mentione equations, we can simulate responses in MATLAB environment. First we consier the saturation element between an 6 accoring to power limitations. Also initial conitions are assume for plate an 5 for the wall. We want to examine ifferent cases in simulation. First suppose η =, λ =, ϕ = 5 an k =. Then check satisfaction of the sliing conition for simulation s as shown in Fig.. Positive values of the plotte curve isplays the fulfillment of sliing conition an therefore the simulation results are vali. U(Electric Power Plate Temp Figure. Sliing conition check Sliing Control for vertical furnace Figure. Step input response with SMC Sliing Control for vertical furnace Figure 5. Electric power behavior with SMC an step input ISSN: Issue 5, Volume, May 9

8 8 Sliing Control for vertical furnace 6 Sliing Control for vertical furnace Plate Temp U(Electric Power 8 6 U(Electric Power Figure 6. Ramp input response with SMC 8 6 Sliing Control for vertical furnace Figure 9. Input signal chattering occurs with ϕ =. Now examine the case k =. In this case the sliing conition is contravene as shown in Fig.. x Sliing Control for vertical furnace Figure 7. Electric power behavior with SMC an ramp input Sliing conition check - Now we change the ϕ = 5 to ϕ =. for investigating the chattering of the input Figure. Sliing conition contravening 9 8 Sliing Control for vertical furnace We can see that in some simulation s, the plotte curve is negative an this means contravening the sliing conition. Plate Temp Sliing Control for vertical furnace Figure 8. Step input response with SMC an selecting ϕ =. Plate Temp ISSN: Issue 5, Volume, May 9

9 U(Electric Power Figure. Step input response with SMC an selecting ϕ =. an λ = Sliing Control for vertical furnace Figure. Input signal chattering occurs with ϕ =. an λ = 9 8 Sliing Control for vertical furnace 7. Conclusion In this paper an inustrial furnace as a nonlinear plant investigate an two ifferent controller esigne for it, as well as a linear system by feeback linearization technique in precise moeling case an sliing moe control in uncertainty of moeling parameters. In feeback linearization control, stability an response spee ajuste by changing k an k in spite of nonlinearity an large ynamic range of the furnace temperature an totally two pairs of poles is selecte an the various outputs behaviors were investigate. In sliing moe control, response spee ajuste by changing the combination of λ, ϕ. Also for softening the control signal an avoiing chattering use an saturation element with large value of ϕ. Parameter η use to ajust the to reaching sliing surface. Increasing the parameter k also helps for retaining satisfying conition. In both of the aforementione controllers, a step isturbance signal applie an the behavior of various parts of the furnace stuie. 7 Plate Temp Figure. Step response of plate temperature with step isturbance an ϕ =. an λ = 6 Sliing Control for vertical furnace References: [] E. Mohseni Languri, D.D. Ganji, N. Jamshii, Variational iteration an Homotopy perturbation methos for fin efficiency of convective straight fins with temperature-epenent thermal conuctivity, 5th WSEAS Int. Conf. on FLUID MECHANICS (FLUIDS'8) Acapulco, Mexico, January 5-7, 8. [] Dr.A. Alizaeh-Attar, H.R. Ghoohestani, I. Nasr Isfahani, Reucing Flare Emissions from Chemical Plants an Refineries Through the Application of Fuzzy Control System, Proceeings of the 6th WSEAS International Conference on Applie Computer Science, Hangzhou, China, April 5-7, 7. U(Electric Power Figure. Input signal behavior with step isturbance an ϕ =. an λ = [] HIMANSHU DEHRA, The Electroynamics of a Pair of PV Moules with Connecte Builing Resistance, Proc. of the r IASME/WSEAS Int. Conf. on Energy, Environment, Ecosystems an Sustainable Development, Agios Nikolaos, Greece, July -6, 7. [] Khajornsak Sopajaree, an Apisit Sancom, CHEMICAL AND PHYSICAL PROPERTY OF RICE STRAW WASTE AND HOSPITAL SEWAGE SLUDGE IN TURNED WINDROW AERATION SYSTEM, r IASME/WSEAS Int. ISSN: Issue 5, Volume, May 9

10 Conf. on Energy & Environment, University of Cambrige, UK, February -5, 8. [5] Archana Thosar, Amit Patra, Souvik Bhattacharyya, Feeback linearization base control of a variable air volume air conitioning system for cooling applications, ISA Transactions, Volume 7, Issue, July 8, Pages 9-9. Systems, Volume 9, Issue, May 7, Pages - 8. [] S. Piñón, E.F. Camacho, B. Kuchen, M. Peña, Constraine preictive control of a greenhouse, Computers an Electronics in Agriculture, Volume 9, Issue, December 5, Pages 7-9. [6] R. Yazanpanah, J. Soltani, G.R. Arab Markaeh, Nonlinear torque an stator flux controller for inuction motor rive base on aaptive input output feeback linearization an sliing moe control, Energy Conversion an Management, Volume 9, Issue, April 8, Pages [7] Enrico De Tuglie, Silvio Marcello Iannone, Francesco Torelli, Feeback-linearization an feeback feeforwar ecentralize control for multimachine power system, Electric Power Systems Research, Volume 78, Issue, March 8, Pages 8-9. [8] Jaho Seo, Raviner Venugopal, Jean-Pierre Kenné, Feeback linearization base control of a rotational hyraulic rive, Control Engineering Practice, Volume 5, Issue, December 7, Pages [9] Cristina M. Cirre, Manuel Berenguel, Loreto Valenzuela, Euaro F. Camacho, Feeback linearization control for a istribute solar collector fiel, Control Engineering Practice, Volume 5, Issue, December 7, Pages 5-5. [] B.B. Sharma, I.N. Kar, Parametric convergence an control of chaotic system using aaptive feeback linearization, Chaos, Solitons & Fractals, In Press, Correcte Proof, Available online 6 October 7. [] Chung-Cheng Chen, Chao-Hsing Hsu, Ying-Jen Chen, Yen-Feng Lin, Disturbance attenuation of nonlinear control systems using an observer-base fuzzy feeback linearization control, Chaos, Solitons & Fractals, Volume, Issue, August 7, Pages [] A. Kazemi, M.R. Jahe Motlagh, A.H. Naghshbany, Application of a new multi-variable feeback linearization metho for improvement of power systems transient stability, International Journal of Electrical Power & Energy ISSN: Issue 5, Volume, May 9