Control Theory & Applications

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1 Control Theory & Applications Optial Dynaic Inversion Control Design for a Class of Nonlinear Distributed Paraeter Systes with Continuous and Discrete Actuators Journal: anuscript ID: anuscript Type: Date Subitted by the Author: Coplete ist of Authors: Keyword: IEE Proc. Control Theory & Applications draft Research Paper n/a Padhi, Radhakant; Indian Institute of Science, Aerospace Engineering Balakrishnan, S.N.; University of issouri-rolla, Aerospace Engineering Optial dynaic inversion, DISTRIBUTED PARAETER SYSTES, TEPERATURE CONTRO

2 Page 1 of 5 Control Theory & Applications Optial Dynaic Inversion Control Design for a Class of Nonlinear Distributed Paraeter Systes with Continuous and Discrete Actuators Radhakant Padhi 1 and S. N. Balakrishnan 1 Departent of Aerospace Engineering, Indian Institute of Science Bangalore, India Departent of echanical and Aerospace Engineering, University of issouri Rolla, USA Abstract Cobining the principles of dynaic inversion and optiization theory, two stabilizing state feedback control design approaches are presented for a class of nonlinear distributed paraeter systes. One approach cobines the dynaic inversion with variational optiization theory and it can be applied when there is a continuous actuator in the spatial doain. This approach has ore theoretical significance in the sense that the convergence of the controller can be proved and it does not lead to any singularity in the control coputation as well. The other approach, which can be applied when there are a nuber of discrete actuators located at distinct places in the spatial doain, cobines dynaic inversion with static optiization theory. This approach has ore relevance in practice, since such a scenario appears naturally in any practical probles because of ipleentation concern. These new techniques can be classified as design-then-approxiate techniques, which are in general ore elegant than the approxiate-thendesign techniques. However, unlike the existing design-then-approxiate techniques, the new techniques presented here do not deand involved atheatics (like infinite diensional operator theory). To deonstrate the potential of the proposed techniques, a real-life teperature control proble for a heat transfer application is solved, first assuing a continuous actuator and then assuing a set of discrete actuators. Keywords: Dynaic inversion, Optial dynaic inversion, Distributed paraeter systes, Teperature control, Design-then-approxiate 1 Asst. Professor, Eail: padhi@aero.iisc.ernet.in, Tel: , Fax: Professor, Eail: bala@ur.edu, Tel: , Fax:

3 Control Theory & Applications Page of 5 1. Introduction There are wide class of probles (e.g. heat transfer, fluid flow, flexible structures etc.) for which a luped paraeter odeling is inadequate and a distributed paraeter syste (DPS) approach is necessary. Control design for distributed paraeter systes is often ore challenging as copared to luped paraeter systes and it has been studied both fro atheatical as well as engineering point of view. An interesting brief historical perspective of the control of such systes is found in [asiecka]. In a broad sense, existing control design techniques for distributed paraeter systes can be attributed to either approxiate-thendesign (ATD) or design-then-approxiate (DTA) categories. An interested reader can refer to [ a Burns] for discussions on the relative erits and liitations of the two approaches. In the ATD approach the idea is to first coe up with a low-diensional reduced (truncated) odel, which retains the doinant odes of the syste. This truncated odel (which is often a finite-diensional luped paraeter odel) is then used to design the controller. One such potential approach, which has becoe fairly popular, first coes up with proble-oriented basis functions using the idea of proper orthogonal decoposition (POD) (through the snapshot solutions ) and then uses those in a Galerkin procedure to coe up with a low-diensional reduced luped paraeter approxiate odel (which usually turns out to be a fairly good approxiation). Out of nuerous literatures published on this topic and its use in control syste design, we cite [Annasway, Arien, Banks, b Burns, Christofides, Holes, Padhi, Ravindran, Singh] for reference. For linear systes, such an approach of designing the POD based basis function leads to the optial representation of the PDE syste in the sense that it captures the axiu energy of the syste with least nuber of basis functions as copared to any other set of orthogonal basis functions [Holes]. For nonlinear systes, however, such a useful result does not exist. Even though the POD based odel reduction idea has been successfully used for nuerous linear and nonlinear DPS in both linear as well as nonlinear probles, there are a few iportant shortcoings in the POD approach: (i) the technique is proble dependent and not generic; (ii) there is no guarantee that the snap-shot solutions will capture all doinant odes of the syste and, ost iportant, (iii) it is usually difficult to have a set of good snap-shot

4 Page 3 of 5 Control Theory & Applications solutions for the closed-loop syste prior to the control design. This is a serious liiting factor for applying this technique in the closed-loop control design. Because of this reason, soe attepts are being ade in recent literature to adaptively redesign the basis functions (and hence the controller) in an iterative anner. An interested reader can see [Annasway, Arien, Ravindran] for a few ideas in this regard. In the DTA approach, on the other hand, the usual procedure is to use infinite diensional operator theory to coe up with the control design in the infinite diensional space first [Curtain]. For ipleentation purpose, this controller is then approxiated to a finite diensional space by truncating an infinite series, reducing the size of feedback gain atrix etc. An iportant advantage of this approach is that it takes into account the full syste dynaics in designing the controller, and hence, usually perfors better [ a Burns]. However, to the best of the knowledge of the authors, these operator theory based DTA approaches are ainly liited to linear distributed paraeter systes [Curtain] and soe liited class of probles like spatially invariant systes [Baeih]. oreover the atheatics of the infinite diensional operator theory is usually involved, which is probably another reason why it has not been able to becoe popular aong practicing engineers. One of the ain contributions of this paper is that it presents two generic control design approaches for a class of nonlinear distributed paraeter systes, which are based on the DTA philosophy. Yet they are fairly straightforward, quite intuitive and reasonably siple, aking it easily accessible to practicing engineers. The only approxiation needed here is rather the spatial grid size selection for the control coputation/ipleentation (which can be quite sall, since the coputational requireents are very inial). In the control design literature for luped paraeter systes, a relatively siple, straightforward and reasonably popular ethod of nonlinear control design is the technique of dynaic inversion (e.g. [Enns], [ane], [Ngo]), which is essentially based on the philosophy of feedback linearization [Slotine]. In this approach, first an appropriate coordinate transforation is carried out to ake the syste dynaics take a linear for (in the transfored coordinates). Then linear control design tools are used to synthesize the controller. Even though the idea sounds elegant, it turns out that this ethod is quite sensitive to odeling and paraeter inaccuracies, which has been a potential liiting factor for its usage in practical applications for 3

5 Control Theory & Applications Page 4 of 5 quite soe tie. However, a lot of research has been carried out in the recent literature to address this critical issue. One way of addressing the proble is to augent the dynaic inversion technique with the H robust control theory [Ngo]. Another way is to augent this control with neural networks (trained online) so that the inversion error is cancelled out for the actual syste ([Ki], [cfarland]). With the availability of these augenting techniques, dynaic inversion has evolved as a potential nonlinear control design technique. Using the fundaental idea of dynaic inversion and cobining it with the variational and static optiization theories [Bryson], two forulations are presented in this paper for designing the control syste for one-diensional control-affine nonlinear distributed paraeter systes. We call this erger as optial dynaic inversion for obvious reasons. Out of the two techniques presented here, one assues a continuous actuator in the spatial doain (we call this as continuous controller ). The other technique assues a nuber of actuators located at discrete locations in the spatial doain (which we call as a discrete controller ). The continuous controller forulation has a better theoretical significance in the sense that the convergence of the controller to its steady-state profile can be proved with the evolution of tie. In the process, unlike the discrete controller forulation, it does not lead to any singularity in the required coputations either. On the other hand, the discrete controller forulation has ore relevance in practice in the sense that such a scenario appears naturally in any (probably all) practical probles (a continuous controller is probably never realizable). To deonstrate the potential of the proposed techniques, a real-life teperature control proble for a heat transfer application is solved, applying both the continuous as well as the discrete control design ideas. A few salient points with respect to the new techniques presented here are as follows. First, even though the optiization idea is used, the new approach is fundaentally different fro optial control theory. The ain driving idea here is rather dynaic inversion, which guarantees stability of the closed loop (the rate of decay of the error rather depends on the selected gain atrix and not on the cost function weights). In addition, this objective is achieved with a iniu control effort (in a weighted or l nor sense), where the cost function plays an iportant role in the sense that it not only leads to a iniu control effort, but also distributes the task aong various available controllers (which are located at different locations in the spatial doain). Second, the technique leads to a state feedback control solution in closed 4

6 Page 5 of 5 Control Theory & Applications for (hence, unlike optial control theory, it does not deand any coputationally intensive procedure in the control coputation). Finally, even though they can be classified into DTA category, the techniques presented do not deand the knowledge of coplex atheatical tools like infinite diensional operator theory. Hence, we hope that the techniques will be quite useful to practicing engineers.. Proble Description.1 Syste Dynaics with Continuous Controller In the continuous controller forulation, we consider the following syste dynaics where the state x( t, y ) and controller (, ) variable y [, ]. x represents x / t (,,,... ) (,,,... ) x = f x x x + g x x x u (1) u t y are continuous functions of tie t and spatial and x, x represent x / y, x / y respectively. We assue that appropriate boundary conditions (e.g. Dirichlet, Neuann etc.) are available to ake the syste dynaics description Eq.(1) coplete. Both x( t, y ) and (, ) u t y are considered to be scalar functions. The control variable appears linearly, and hence, the syste dynaics is in the control affine for. Furtherore, we assue that the function g( x, x, x,... ) fro zero, i.e. ( ) is bounded away g x, x, x,... t, y. In this paper, we do not take into account those situations where control action enters the syste dynaics through the boundary actions (i.e. boundary control probles are not considered).. Syste Dynaics with Discrete Controllers In the discrete controller forulation, we assue that a set of discrete controllers u are located at y ( = 1,, ) locations, with the following assuptions: The width of the action of the controller located at y is w. 5

7 Control Theory & Applications Page 6 of 5 In the interval [ y w /, y + w /] [, ], the controller u (, ) t y is assued to have a constant agnitude. Outside this interval, u =. However, the interval w ay or ay not be sall. There is no overlapping of the controller located at y with its neighboring controllers. No controller is placed exactly at the boundary, i.e. the control action does not affect the syste through boundary actions. For this case the syste dynaics can be written as follows = (,,, ) + (,,, ) x f x x x g x x x u () = 1.3 Goal for the Controller The goal for the controller in both continuous and discrete actuator cases is sae; i.e. the controller should ake sure that the state variable x( t, y) x ( t, y) as t for all y [, ] where x ( t, y ) is a known (possibly tie-varying) profile in the doain [ ] continuous in y and satisfies the spatial boundary conditions.,,, which is 3. Synthesis of the Controllers 3.1 Synthesis of Continuous Controller First, we define an output (an integral error) ter as follows 1 z() t = x( t, y) x ( t, y) dy (3) Note that when z() t, x( t, y) x ( t, y) everywhere in y [, ]. Next, following the principle of dynaic inversion [Enns, ane, Ngo, Slotine], we attept to design a controller such that the following stable first-order equation (in tie) is satisfied 6

8 Page 7 of 5 Control Theory & Applications z + k z = (4) where, k > serves as a gain; an appropriate value of k has to be chosen by the control designer. To have a better physical interpretation, one ay choose k ( 1/ τ ) =, where τ > serves as a tie constant for the error z( t ) to decay. Using the definition of z fro Eq.(3), Eq.(4) leads to k x x x x dy x x dy (5) ( )( ) = ( ) Substituting for x fro Eq.(1) in Eq.(5) and siplifying we arrive at ( ) (,,,... ) x x g x x x u dy = γ k where γ ( x x ) f ( x, x, x,... ) x dy ( x x ) dy (6) Note that the value for u( t, y ) satisfying Eq.(6) will eventually guarantee that z( t) as t. However, since Eq.(6) is in the for of an integral, there is no unique solution can be obtained for u( t, y ) fro it. To obtain a unique solution, however, we have the freedo of putting an additional goal. We take advantage of this fact and ai to obtain a solution for u( t, y ) that will not only satisfy Eq.(6), but at the sae tie, will also iniize the cost function 1 J = r( y) u( t, y) dy (7) In other words, we wish to iniize the cost function in Eq.(7), subjected to the constraint in Eq.(6). An iplication of choosing this cost function is that the ai is to obtain the control solution u( t, y ) that will lead to x( t, y) x ( t, y) ( ), [, ] with iniu control effort. In Eq.(7), r y > y is the weighting function, which needs to be chosen by the control designer. This weighting function gives the designer the flexibility of putting relative iportance of the control agnitude at different spatial locations. Note that the choice of 7

9 Control Theory & Applications Page 8 of 5 + ( ) = [, ] r y c R y eans the control agnitude is given equal iportance at all spatial locations. Following the technique for constrained optiization [Bryson], we first forulate the following augented cost function 1 J = ru dy+ x x gudy λ ( ) γ (8) where λ is a agrange ultiplier, which is a free variable needed to convert the constrained optiization proble to a free optiization proble. In Eq.(8), we have two free variables, naely u and λ. We have to iniize J by appropriate selection of these variables. The necessary condition of optiality is given by [Bryson] δ J = (9) where δ J represents the first variation of J. However, we know that δ J = ru δu dy + λ x x g δu dy + δλ x x g u dy γ [ ] ( ) ( ) ( ) ( ) = ru λ x x g δu dy δλ x x g u dy γ + + (1) Fro Eqs.(9) and (1), we obtain ( ) ( ) ru + λ x x g δu dy + δλ x x g u dy γ = (11) Since Eq.(11) ust be satisfied for all variations δu satisfied siultaneously ( ) and δλ, the followings equations should be ru + λ x x g = (1) ( ) x x g u dy = γ (13) 8

10 Page 9 of 5 Control Theory & Applications Note that Eq.(13) is nothing but Eq.(6a). Solving for u fro Eq.(1) we get ( λ / )( ) u = r x x g (14) Substituting the above expression for u in Eq.(13) and solving for λ we get λ = γ ( ) x x g dy r (15) Substituting this expression for λ back in Eq.(14), we finally obtain u = ( ) r y ( ) ( ) γ x x g x x g ( ) r y dy (16) As a special case, if r( y) locations) and g( x, x, x,... ) + = c (i.e. equal weightage is given to the controller at all spatial = β R, then Eq.(16) siplifies to ( x x ) ( ) γ u = β x x dy (17) It ay be noticed that when x( t, y) x ( t, y) = (i.e. perfect tracking occurs), there is soe coputational difficulty in the sense that a zero sees to appear in the denoinator of Eqs.(16-17), which leads to singularity in the control solution u, i.e. u. However, even though this sees to be obvious, it does not happen. To see this, we will show that when x( t, y) x ( t, y), (, ) (, ), where u ( t, y ) is defined as the control required to keep x( t, y ) at x ( t, y ) u t y u t y (see Eq.(19)). Before showing this, however, we need a non-trivial expression for u (, ) that, when x( t, y) x ( t, y), y [, ], fro Eq.(1) that we can write t y. For 9

11 Control Theory & Applications Page 1 of 5 x = f + g u ( ) ( ) where f f x, x, x,, g g x, x, x, (18) Fro Eq.(18), we can write the control solution as Note that the solution u (, ) class of DPS considered here, g ( ) g x, x, x, 1 u ( t, y) = f x g (19) t y in Eq.(19) will always be of finite agnitude, since for the = is always bounded away fro zero. Also note that in actual ipleentation of the controller, we ay rarely encounter the condition (, ) (, ) [, ] x t y = x t y y, since it is very difficult to eet. However, this expression is useful in the convergence analysis of the controller. Next, we state and prove the following convergence result. Theore u( t, y ) in Eq.(16) converges to u ( t, y ) in Eq.(19) when x( t, y) x ( t, y) y [, ] Proof: First we notice that at any point y ( ),., the control solution in Eq.(16) can be written as ( ) u y k x( y) x ( y ) g( y) x( y) x ( y) f ( y) x ( y) dy x( y) x ( y) dy + = x y x y g y r y dy ( ) ( ) ( ) ( ) r( y) () We want to analyze this solution for the case when x( t, y) = x ( t, y) for all y [, ] Without loss of generality, we analyze the case in the liit when x( t, y) x ( t, y) [ ε /, + ε /] [, ], ε and x( t, y) x ( t, y) y y y liiting case, let us denote u( t, y ) as u( t, y ), which is given by., for = everywhere else. In such a 1

12 Page 11 of 5 Control Theory & Applications (, ) u t y ε ε y + k ( ) ( ) ( ) y + x ty, x ty, gty, ε xty (, ) x ( ty, ) f ( ty, ) x ( ty, ) dy+ ε xty (, ) x ( ty, ) dy y y = ε y + xty (, ) x( ty, ) gty (, ) r( y) ε dy y r( y) xty (, ) x ( ty, ) k gty (, ) xty (, ) x ( ty, ) f ( ty, ) x ( t, y ) ε + x( t, y) x ( t, y) ε = xty (, ) x( ty, ) gty (, ) r( y ) ε r y 1 = f t y x t y g t y = u (, ) ( t, y ) (, ) (, ) oreover, this happens y ( ). Hence u( t, y) u ( t, y) as x( t, y) x ( t, y), y [, ], This copletes the proof. Final Control Solution for Ipleentation Cobining the results in Eqs.(16) and (17), we finally write the control solution as ( ) (1). u 1 f x, if x( t, y ) x ( t, y) y [, ] g = γ ( x x ) g =, otherwise ( x x ) g r( y) dy r( y) () Even though u( t, y) u ( t, y) when x( t, y) x ( t, y) y [, ], in the nuerical ipleentation of the controller, it is advisable to exercise the caution as outlined in Eq.() to avoid nuerical probles in coputer prograing. One can notice in the developent of Eq.() that there was no need of approxiating the syste dynaics to coe up with the closed for control solution. However, to copute/ipleent the control, there is a requireent for choosing a suitable grid in the spatial doain. Hence, the technique proposed can be classified into the design-then-approxiate category. Note that a finer grid can be selected to copute u (, ) t y since the only coputation that depends on the grid size in Eq.() is a nuerical integration, which does not deand intensive coputations. 11

13 Control Theory & Applications Page 1 of 5 3. Synthesis of Discrete Controllers In this section we concentrate on the case when we have only a set of discrete controllers (as described in Section.). In such a case, following the developent in continuous forulation (Section 3.1), we arrive at the following equation ( ) (,,,... ) (, ) x x g x x x u y w dy = γ (3) = 1 where γ is as defined in Eq.(6b). Expanding Eq.(3), we can write w1 w y1 + y + w ( x x ) g dy ( ) 1 u1 + + w x x g dy u = γ y1 (4) y For convenience, we define w y + w y ( ) I x x g dy, = 1,, (5) Then fro Eqs.(4) and (5), we can write Iu+ + I u = γ (6) 1 1 Eq.(6) will eventually guarantee that z( t) as t. However, note that Eq.(6) is a single equation with variables u, = 1,, and hence we have infinitely any solutions. To obtain a unique solution, we ai to obtain a solution that will not only satisfy Eq.(6), but at the sae tie will also iniize the following cost function 1 J = ( r1w1u1 + + rwu ) (7) In other words, we wish to iniize the cost function in Eq.(7), subjected to the constraint in Eq.(6). An iplication choosing this cost function is that we wish to obtain the solution that will lead to iniu control effort. In Eq.(7), choosing appropriate values for r 1,, r > 1

14 Page 13 of 5 Control Theory & Applications gives a control designer the flexibility of putting relative iportance of the control agnitude at different spatial locations y, = 1,,. Following the principle of constrained optiization [Bryson], we first forulate the following augented cost function 1 J = ( r1w1u1 + + rwu ) + λ ( I1u Iu ) γ (8) where λ is a agrange ultiplier, which is a free variable needed to convert the constrained optiization proble to a free optiization proble. In Eq.(8) we have λ and u, 1,, = as free variables, with respect to which the iniization has to be carried out. The necessary condition of optiality [Bryson] leads to the following equations J u =, = 1,, (9) J = λ (3) Expanding Eqs.(9) and (3) leads to r w u + I λ =, = 1,, (31) Iu+ + I u = γ (3) 1 1 Solving for u, 1 u fro Eq.(31), substituting those in Eq.(3) and solving for λ we get, λ = = 1 I γ / ( rw ) (33) Eqs.(31) and (33) lead to the following expression 13

15 Control Theory & Applications Page 14 of 5 u I γ = rw I rw / = 1 ( ), = 1,, (34) As a special case, when r 1 = = r (i.e. equal iportant to iniization of all controllers) and w = = w (i.e. widths of all controllers are sae), we have 1 u I γ = (35) I I I1 I where [ ] T. Note that in case we have a nuber of controllers being applied over different control application widths (i.e. u, = 1,, are different), we can still use the siplified forula in Eq.(35), if it leads to satisfactory syste response by choosing r 1, such that rw 1 1 = = r w., r Singularity in Control Solution and Revised Goal: Fro Eqs.(34) and (35), it is clear that when I (which happens when all of I,, 1 I ) and γ, there is a proble of singularity in the control coputation in the sense that u (this happens since the denoinators of Eqs.(34-35) go to zero faster than the corresponding nuerators). Note that if the nuber of controllers is large, probably the occurrence of such a singularity is a rare possibility, since all of I 1,, I siultaneously is rather a strong condition. Nevertheless such a case ay arise during transition. ore iportant, this issue of control singularity will always arise when x( t, y) x ( t, y), y [, ] (which is the priary goal of the control design). This happens possibly because we have only liited control authority (controllers are available only in a subset of the spatial doain), whereas we have aied to achieve a uch bigger goal of tracking the state profile y [, ] - soething that is beyond the capability of the controllers. Hence whenever such a case arises (i.e. when all of I,, 1 I or, equivalently, I ), to avoid the issue of control singularity, we propose to redefine the goal as follows. 14

16 Page 15 of 5 Control Theory & Applications,, T X x1 x, First, we define [ ] T and the error vector ( E X X ) X x,, 1 x. Next, we ai to design a controller such that E as t. In other words, we ai to guarantee that the values of the state variable at the node points ( y, = 1,, ) track their corresponding desired values. We do this, we select a positive definite gain atrix K such that: E + K E = (36) One way of selecting such a gain atrix K is to choose it a diagonal atrix with eleent being k ( 1/ τ ) such a case, the th diagonal = where τ > is the desired tie constant of the error dynaics. In th channel of Eq.(36) can be written as e + k e = (37) Expanding the expressions for e and e and solving for u ( = 1,, ), we obtain 1 u = x f k ( x x ) g (38a) where x x( t, y ), x x ( t, y ), f f ( t, y ), g g( t, y ) (38b) Final Control Solution for Ipleentation Cobining the results in Eqs.(34) and (38), we finally write the control solution as u 1 x f k( x x ), if I tol g < = I γ, otherwise rw I / ( rw ) = 1 (39) where tol represents a tolerance value. An appropriate value for this tuning variable can be fixed by the control designer. Note that soe discontinuity/jup in the control agnitude is expected when the switching takes place. However, this jup can be iniized by judiciously selecting a proper tolerance value. 15

17 Control Theory & Applications Page 16 of 5 One can notice that there was no need of approxiating the syste dynaics (like reducing it to a low-order luped paraeter odel) to coe up with the closed for control solution in Eq.(39). However, like the continuous controller forulation, to copute/ipleent the control, there is a requireent for choosing a suitable grid in the spatial doain. Hence, this technique can also be classified into the design-then-approxiate category. In this case too, a finer grid can be selected to copute u, = 1,, since the only coputation that depends on the grid size in Eq.(39) is a series of nuerical integrations, which do not deand intensive coputations. 4. A otivating Nonlinear Proble 4.1 atheatical odel The proble used to deonstrate the theories presented in Section 3 is a real-life proble. It involves the heat transfer in a fin of a heat exchanger, as depicted in Figure 1. Figure 1: Pictorial representation of the physics of the proble First we develop a atheatical odel fro the first principles of heat transfer [iller]. Using the law of conservation of energy in an infinitesial volue at a distance y having length y, we write 16

18 Page 17 of 5 Control Theory & Applications Q + Q = Q + Q + Q + Q (4) y gen y+ y conv rad chg where Q y is the rate of heat conducted in, heat conducted out, Q gen is the rate of heat generated, Q y + y is the rate of Q conv is the rate of heat convected out, Q rad is the rate of heat radiated out and Q chg is the rate of heat change. Next, fro the laws of physics for heat transfer [iller], we can write the following expressions Q y T = ka y (41a) Qgen = S A y (41b) ( ) = (41c) Qconv h P y T T 1 rad 4 4 ( ) Q = εσ P y T T (41d) Q chg T = ρ C A y t (41e) In Eqs.(41a-e), T( t, y ) represents the teperature (this is the state (, ) x t y in the context of discussion in Section 3), which is a function of both tie t and spatial location y. S( t, y ) is the rate of heat generation per unit volue (this is the control u in the context of discussion in Section 3) for this proble. The eanings of various paraeters and their nuerical values used and are given in Table 1. Table 1: Definitions and nuerical values of the paraeters Paraeter eaning Nuerical value k o Theral conductivity 18 W / ( C ) A Cross sectional area c P Perieter 9 c h Convective heat transfer coefficient 5 W /( C ) T Teperature of the ediu in the 1 iediate surrounding of the surface 3 C 17

19 Control Theory & Applications Page 18 of 5 T Teperature at a far away place in the direction noral to the surface ε Eissivity of the aterial. 4 C σ 8 4 Stefan-Boltzann constant W / K ρ Density of the aterial 3 7 kg / C Specific heat of the aterial 86 J / ( kg C) The values representing of the properties of the aterial were chosen assuing Aluinu. The area A and perieter P have been coputed assuing the a fin of diension 4c 4c.5c. Note that we have ade a one-diensional approxiation for the dynaics, assuing unifor teperature in the other two diensions being arrived at instantaneously. Using Taylor series expansion and considering a sall y, we can write Q + y y Qy+ y Qy y (4) Using Eqs.(41a-e) and (4) in Eq.(4) and siplifying, we can write T k T P = ht ( T ) + εσ( T T ) S 1 + t ρc y AρC ρc (43) For convenience, we define α ( k ρc), α ( Ph) ( AρC), α ( Pεσ ) ( AρC) ( ρ ) β 1/ C, we can rewrite Eq.(43) as 1 / / and 3 / T t T 4 4 = α1 + α( T T ) + α ( ) 1 3 T T + β S y (44) Along with Eq.(44), we consider the following boundary conditions T y= T = Tw, = y y= (45) 18

20 Page 19 of 5 Control Theory & Applications where T w is the wall teperature. We have assued insulated boundary condition at the tip with the assuption that either there is soe physical insulation at the tip or the heat loss at the tip due to convection and radiation is negligible (ainly because of its low surface area). The goal for the controller was to ake sure that the actual teperature profile T( t, y) T ( y) we chose T ( y ) to be a constant (with respect to tie) teperature profile. T ( ) generated by using the following expression ( ) w ( w tip) T y T T T ζ y, where y was = + (46) In Eq.(46) we chose the wall teperature decaying paraeter Tw ζ =. The selection such a T ( ) = 15 C, fin tip teperature Ttip = 13 C and the y fro Eq.(46) was otivated by the fact that it leads to a sooth continuous teperature profile across the spatial diension y. This selection of T ( y ) satisfies the boundary condition at y = exactly and at y = approxiately, with a very sall (rather negligible) approxiation error. Note that the syste dynaics is in control-affine for and g( x x x ),,, = β. oreover, there is no boundary control action. This is copatible with the class of DPS for which we have developed the control synthesis theories in Section 3. In the discrete controller case, the syste dynaics in Eq.(44) will get odified to T t T 4 4 = α1 + α( T T ) + α ( ) 1 3 T T + β S y (47) = 1 However, the boundary conditions reain sae as in Eq.(45). 4. Synthesis of Continuous Controller In our siulation studies with the continuous controller forulation, we selected the control gain as k = 1/ τ, where 3 sec τ =. We assued ( ) + r y as a constant c, and hence, were able to use the siplified forula for the control in Eq.(17). Hence a nuerical value for r( y ) was not necessary for the siulation studies. 19

21 Control Theory & Applications Page of 5 First we chose an initial condition (profile) for the teperature as obtained fro the expression T(, y) T x(, y) teperature and x(, ) = +, where T = 15 C (a constant value) serves as the ean y represents the deviation fro T. Taking 5 as x(, y) ( A/ ) ( A/) cos( π πy/ ) A = we coputed x(, y ) = + +. Applying the controller as synthesized in Eq.(), we siulated the syste in Eqs.(44)-(45) fro tie t = t = to t = t f = 5in. The results obtained are as in Figure (a,b). We can see fro Figure (a) that the goal of tracking T ( y ) is et without any proble. The associated control (rate of energy input) profile S( t, y ) obtained is as shown in Figure (b). It is iportant to note that even as T( t, y) T ( y), there is no control singularity. In fact the control profile develops (converges) towards the steady-state control profile (see Eq.(19)). Figure (a): Evolution of the teperature (state) profile fro a sinusoidal initial condition Figure (b): Rate of energy input (control) for the evolution of teperature profile in Figure (a) Next, to deonstrate that siilar results will be obtained for any arbitrary initial condition of the teperature profile T(, ) T (, ) y, we considered a nuber of rando profiles for y and carried out the siulation studies. The rando profiles using the relationship (, ) = + (, ), where x(, ) T y T x y y was generated using the concept of Fourier Series, such that it satisfies x(, y) k x, x (, y) k x and x (, y) k x 1 ax ax 3 ax. The values for x, x and x were coputed using an envelope profile x ( ) sin ( / ) ax ax ax env y = A π y. The

22 Page 1 of 5 Control Theory & Applications ( ) nor used is the nor defined by ( ) x x y dy 1/. We selected the value of paraeter A as 5 and selected k 1 =, k = k3 = 1. For ore details about the philosophy of generation of these rando profiles, the reader is referred to [Padhi]. The results obtained fro such a rando initial condition are as in Figure 3(a,b). Once again, we clearly notice that the objective of (, ) ( ) T t y T y is et. We also notice that the control (rate of energy input) agnitude is not high and, ore iportant, the control profile develops towards and converges to the steady-state control profile as coputed fro Eq.(19). Figure 3(a): Evolution of the teperature (state) profile fro a rando initial condition Figure 3(b): Rate of energy input (control) for the evolution of teperature profile in Figure 3(a) 4.3 Synthesis of Discrete Controllers In our siulation studies with the discrete controller forulation, we selected the control gain as k ( 1/ τ ) =, whereτ = 3 sec. While checking condition to switch the controller, the tolerance value was selected as tol =.1. After switching, we used the control gain ( ) K = diag k1 k and selected k = 1/ τ, τ = τ for = 1,,. We took w1 = = w = c and assued r 1 = = r. Because of this there was no need to select nuerical values for r 1, r. To begin with we selected = 5 (five controllers), located at equal spacing., 1

23 Control Theory & Applications Page of 5 First we chose the sae sinusoidal initial condition (profile) for the teperature as used for the continuous controller forulation. Applying the controller as synthesized in Eq.(39), we siulated the syste odel in Eqs.(47) and (45) fro tie t = t = to t = t f = 5in. The results obtained are as in Figure 4(a,b). We can see fro Figure 4(a) that the goal of tracking T ( ) y is roughly et. The associated control (rate of energy input) profile S( t, y ) obtained is as shown in Figure 13. The figure shows that the required control agnitude is not very high in the entire, and for all tie t t, t f. Note that as copared to the continuous case, the control effectiveness is saller in the discrete case (control is applied only at a sall subspace of the entire spatial doain). However, since we aied the sae decaying rate for the state error, in tune with the intuition one can observe that the agnitude of the discrete controllers are higher as copared to the continuous forulation (see Figures (b) and 4(b)). spatial doain [ ] Figure 4(a): Evolution of the teperature (state) profile fro a sinusoidal initial condition Figure 4(b): Rate of energy inputs (controllers) for the evolution of teperature profile in Figure 4(a) We notice a few sall probles in the results in Figures 4(a,b). First, there are sall jups in the control histories when the control switching takes place (at about.5 in). oreover, we see soe weaving nature of the state profile as T( t, y) T ( y), and hence, the goal for control design is not et to a satisfactory level. Both of these probably happened because we assued a sall nuber of discrete controllers. One way of iniizing this effect is

24 Page 3 of 5 Control Theory & Applications to increase the nuber of controllers. Next, we selected ten controllers (instead of five) and carried out the siulation again. The results are shown in Figure 5(a,b). It is quite clear fro this figure that the weaving nature is substantially saller and the goal T( t, y) T ( y), y [, ] is et with ore accuracy. Also note that as copared to the case with five controllers, here the control effectiveness is higher and subsequently the agnitudes of the controllers are saller (copare Figures 4(b) and 5(b)). Figure 5(a): Evolution of the teperature (state) profile fro a sinusoidal initial condition Figure 5(b): Rate of energy inputs (controllers) for the evolution of teperature profile in Figure 5(a) To deonstrate that siilar results will be obtained for any arbitrary initial condition of the teperature profile T(, y ), next we considered a nuber of rando profiles for T(, y ) (generated the sae way as in Section 4.) and carried out the siulation studies. The results obtained fro such a rando initial condition are quite satisfactory in the sense that the tracking objective was et. To contain the length of the paper, however, we do not include those results. 5. Conclusions Based on the newly proposed optial dynaic inversion theory, two stabilizing state feedback control design approaches are presented for a class of nonlinear distributed paraeter systes. One approach cobines the dynaic inversion with variational optiization, whereas 3

25 Control Theory & Applications Page 4 of 5 the other one (which is ore relevant in practice) can be applied when there are a nuber of discrete actuators located at distinct places in the spatial doain. These new techniques can be classified as design-then-approxiate ethods, which are in general ore elegant than the approxiate-then-design ethods. The forulation leads to a closed for control solution, and hence, is not coputationally intensive. To deonstrate the potential of the proposed techniques, a real-life teperature control proble for a heat transfer application is solved, first assuing a continuous actuator and then assuing a set of discrete actuators and proising nuerical results are obtained. References 1. Annasway A., Choi J. J., Sahoo D., Active Closed oop Control of Supersonic Ipinging Jet Flows Using POD odels, Proceedings of the 41 st IEEE Conference on Decision and Control, as Vegas,.. Arian E., Fahl. and Sachs E. W., Trust-region Proper Orthogonal Decoposition for Flow Control, NASA/CR--114, ICASE Report No Baeih B., The Structure of Optial Controllers of Spatially-invariant Distributed Paraeter Systes. Proceedings of the Conference on Decision & Control, 1997, Banks H. T., Rosario R. C. H and Sith R. C., Reduced-Order odel Feedback Control Design: Nuerical Ipleentation in a Thin Shell odel, IEEE Transactions on Autoatic Control, Vol. 45,, Bryson A. E. and Ho Y. C., Applied Optial Control, ondon: Taylor and Francis, a Burns J.A. and King, B.B., Optial sensor location for robust control of distributed paraeter systes. Proceedings of the Conference on Decision and Control, 1994, b Burns J. and King B. B., A Reduced Basis Approach to the Design of ow-order Feedback Controllers for Nonlinear Continuous Systes, Journal of Vibration and Control, Vol.4, 1998, Christofides P. D., Nonlinear and Robust Control of PDE Systes ethods and Applications to Transport-Reaction Processes, Birkhauser, Boston,. 4

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