180 IEEE TRANSACTIONS ON AUTOMATIC CONTFtOI VOL AC 24 NO 2 APRIL 1979. and demonstrated the workability of the state estimation. algorithm In the present paper we explore the properties. ST NLESSSTEEL 303, HOLLOWCYLINDER, of the state estimation scheme in much more detail and. extend the study to include both deterministic and. COOLING WATEROUT, stochastic feedback control, 11 THEPROCESS Fig 1 Axial cross section of the experimental apparatus. The apparatus used in our experiments is the same as. described in 4 and in more detail in 5 The process where A are solutions to certain transcendental equa. consists of a stainless steel cylinder with a hole drilled tions 4 5 Inserting 6 into 1 4 yields a set of N M. through the center to allow the passage of cooling water equations in the eigencoefficients. A three zone furnace permits the control of both radial U Au B u t 9. and axial temperature profiles Fig 1 shows an axial cross. section of the apparatus Thermocouples are placed at where A is the diagonal matrix comprised of the eigenval. four radial positions r i 1 4 in eight axial planes ues of the system while BD is an N M X 3 matrix and. located at z k 1, 8 In dimensionless form the equa given by. tions governing this system can be written as n 1 2 N. B D bq m 1 2 M 10, a r ar a az S r, I gT z u t i 1 2 3. ro r l O z l t O 1 1, ae brim n l J g z rF z dz 1 1. 0 at z O Z 1, aZ The spatially discrete temperature measurements are. BiQ at r r o given by, Yjk t 6 rj z r qj t i 1 2 N k 1 2 MZ. U Ao Bu 5 where 8 rj z t is the true dimensionless temperature at. location r rj and z z and qik t represents the, where 8 r z t is the dimensionless temperature within the measurement error. cylinder u t is the dmensionless amplitude of the heat. flux of the heaters and u t represents the power input. 111 STATE ESTIMATION, into the heater The parameter Bi is a biot number for. heat transfer A and B are diagonal matrices describing The optimal state estimation equations are derived cf. the heater dynamics The vector g z is the spatial distrib 4 5 such that for the present study the following least. ution function for the furnace heaters Equations 1 4 squares performance index is minimized. may be solved by modal approximation using the eigen. function expansion, r z E E am t n r m z, where N and M are the total number of radial and axial. eigenfunctions Qn r and Gm z used in the approximation. and anm t is their amplitude Asshown in 4 5 the, eigenfunctions of the system are. where f r z t is the right hand side RHS of 1 and hrr. and and M denote the maximal number of radial and axial. measurement positions used by the estimator The spatial. integral minimizes the model error while the second term. minimizes the measurement error The quantity Qvkl t is. LAUSTERER AND RAY STATE ESTIMATION AND FEEDBACKCONTROL 181. a positive definite weighting function and R is defined by. 1 1 r p z z d f N, z N z M M N M, where R r p z t is a positive definite symmetric. weighting function The filter equations take the form 4. a l a a 6 r l g z u t where, at ar2 r ar az2, are the eigenvalues of the covariance equation 5 and. r mp f SISIJISIR r r z z t a r, p r I m z I p rr drdr dzdz 23. Here B r z t denotes the filtered estimate of B r z t and. the covariance P r r z z t of this estimate is given as In addition we have assumed Qvk t to be diagonal and. the solution of time independent i e Qijkl f Qik N and M are the. total numbers of radial and axial eigenfunctions used in. the covariance equations With two radial and seven axial. eigenfunctions the standard case for our experiments. I ap 20 represents 14 coupled linear equations while 21. wouldyield a total of 196 coupled nonlinear equations. N N M M Due to the symmetry of P r r z z t however these. reduce to 84equations still a formidable system Note. that 20 and 21 are decoupled because of the linearity. P r r z z t R r r z z t of the model and measurement equations. Before these equations can be solved numerically one. with the boundary conditions mustspecify proper initial conditions B r z 0 and. P r r z z O according to best a priori estimates How. atz Oandz l ever in our experiments we often chose a rather poor. initial condition for the filter in order to investigate its. ap convergence properties, Bi P at r r o, 0 at r l IV OPTIMAL FEEDBACK CONTROL. and similar conditions at z O z 1 r ro and r 1 It In a similar fashion optimal controller equations are. is assumed that the state vector u t of the heaters is given derived which minimize the quadratic cost functional. by 5 and need not be estimated By application of the. eigenfunction expansion the optimal filter and covariance. equations read, e r z fr B r z t dr dr dz dz, yd r r z z t Bd r z t B r z t drdr dzdz. u t u t r t u t u t df 24, where BAr z t is the desired temperature profile at time t. The quantity P is a true estimate covariance only in the case where. and u t is the corresponding control setting yf r r z z. the process noise and measurement errors are Gaussian and white with and yd r r z z t are positivesemidefiniteweighting. covariances R and QG respectively Forother types of noise functions of the deviations of the state variables from the. processes P must be considered in general to be only a filter parameter. without precise statistical significance desired state while rJt is a positive definite matrix. 182 IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL AC 24 NO 2 APRIL 1979. weighting the control effort The optimal control law takes RICCATI EQUATIONS. V SUBOPTIMAL, A closer look at 21 and 28 reveals that if the initial. conditions and weighting functions are chosen to be diag. onal the only coupling between the equations is due to. Bd r t t e T Z t g z 6 r 1 drdr dtdZ 25 the quadratic terms This suggests a way to simplify the. coupled system of Riccati equations byneglecting any. where R is the solution of off diagonal terms in the quadratic expansion In this. case the suboptimal Riccati equations take the form. parametric changes can be made directly without requir. ing laborious off line calculation for each parameter set. One must stress here however that the decoupling is only. feasiblebecause in our system as our computations, showed the diagonal terms are dominant and the actual. coupling is very weak It is not expected that such subop. 26 timal procedures would be a good approximation in gen. eral unless the system has diagonal dominance, The suboptimal filter equations and suboptimal feed. back controller now take the simpler form, so that there is considerable savings in on line computa. k m z p z 29, N R and MR denote the number of radial and axial eigen. functions used in the controller Riccati equations 28 In VI CHOICE FUNCTIONS. OF THE WEIGHTING, formulating the optimal linear quadratic control law 25. we have neglected the heater dynamics and assume the Before the algorithms developed above can be imple. desired heat flux isimposed instantaneously Forour mented for real time estimation and control the weighting. small laboratory systemthis introduces some error into functions to be used in the quadratic objectives have to be. the feedback control scheme due to the implemented heat specified appropriately It turns out that this is the crucial. flux lagging the theoretical value however for a large step determining the dynamic behavior of the estimator. scale industrial system the heater dynamics would be and controller For a distributed parameter system one. negligible because gas fired furnaces respond much faster must choose not only the time dependence but also the. than electrical heaters spatial shape of these functions To make the best choice. LAUSTERER AND RAY STATE ESTIMATION AND FEEDBACK CONTROL 183. of these weighting functions one must build in as much cooled inner wall it makes sense to choose the simplest. physical knowledge as possible about the system and the radiallyincreasing function in 38 In addition P r r. goals of the control system Withthistype of physical must be symmetricin r and r since P is a symmetric. insight we found that the first set of filter and controller function This choice is also of particular mathematical. parameters selected usually gave good results This is in convenience permitting analytical solutions of the in. contrast to reports of parameter tuning difficulties where tegrals resultingfrom the eigenfunction expansion f z z. early lumping was done i e when one destroys the in 37 was chosen to reflect the error correlation in the. distributed character of theproblemby approximate axial direction which should be symmetric in z and z and. lumping in the first instance followedby direct applica a decreasing function of I Z Z Since the eigenfunctions. tion of lumped parameter control theory Thus there is are cosines it makessomesense to choose a cosine. great valuein retaining the distributed nature of the correlation but many other functions could have been. problem in order to allow the physics to aid in parameter chosen as well To investigate the influence of assuming. selection Our parameter selection procedure is described axial error correlation we also considered. in what follows, For a linear system with Gaussian white noise processes f z z 6 z z 40. as described above it can be shown that which states that the estimation error is axially uncorre. P o E o B o lated, The process noisewhich we implement in a discrete. e r z 0 B r I 0 34 way was selected to take one of two forms. whichis the covariance of the initial estimation error 1. Tidcos z z, R r r z z nAt dis 41, Using this idealized Gaussian white noise case as a guide. choosing our filter parameters we can say that or, R r r z z t should be proportional to the modeling. error covariance while Qvkl t is to be choseninversely. proportional to the measurement error covariance, Note that we assumed in both cases that the error is not. One should note thatour theoretical formulation, assumes a continuous processwith continuous measure correlated radially which can be justified by the small. ments whilein implementation we takeonly discrete ratio of radius to length L 2 lop2 Theconstants Si. and T20d the mean square amplitude of the errors can. measurements with sampling interval At and implement. be chosen reasonably well from physical considerations. control at control interval At Thus the continuous noise. parameters can be approximated in the following way 5 and then improved according to practical experience. If we assume time independence and no spatial correla. R r r z t R f z n t disAts 35 tion of the measurement error whichisgenerally reason. able we can define the discrete measurement weights as. QG t Qs nAt At 36, where R r r z z nAt dis Qi nAt are discrete pro. cess noise and measurement error covariances to be de. fined below A third filter parameter is the initial error where a d. denoteKronecker symbols and is, covariance which we assume to take the form the variance of the measurement error. Let us now define the controller parameters Recogniz. P r r z z O S02f r r fz z z 37 ing that each heater section is almost identical andthat. off diagonal elements in the weighting matrix I would. where Si is a constant the mean squared initial error and. have no physical significance we define, f f are spatial functions which we chose as. r r rr 38 r k I 44, and where I is a 3 X 3 identity matrix and k is a constant. gain factor to be chosen such that fie controllers just do. fz z z cos z Z, 1 39 not saturate, For the state variable weighting function in the feed. These particular functions were chosen according to the back controller objective 241 we chose. followingphysical considerations In mostcasesin our. experiments the filter receives an erroneously low and yd r r z Z t y r r y Z z 45. spatially uniform initial condition equal to the cooling. water temperature Since the radial temperature gradient. ne of thepractical limitations in testing optimal hearquadratic. in the ingot is positive because the feedback controllers is to choose control weights the controllers. heat flux is directed from the heated outer wall to the remain unconstrained. 184 EEE TRANSACTIONS ON AUTOMATIC CONTROL VOL AC 24 NO 2 APRIL 1979. where yr y are selected as follows Because we want to 3 SN suboptimal filter with no axial correlation. steer our system as close to the desired state as possible weighting parameters given by 40 42. with no axial preference and there is no physical reason 4 OC full optimal feedback controller. to correlate the deviation at r z with the deviation at 5 SC suboptimal feedback controller neglecting the. r z we set off diagonal terms in the Riccati equations. 6 FS fixed step manual control no feedback, yr r i rS t r 46 The number of thermocouples chosen was either 1 3 or 8. y 2 z z S 2 2 47 depending on the number of external surface ingot tem. peratures provided to the filter The particular sensors. Note that the factor r in 46 was introduced not only for provided are marked on each of the figures below. mathematical convenience but also to reflect the fact that In order to be able to present the multitude of experi. the filter receivesonly surface measurements Thus we ments in condensed form and to facilitate comparison a. give more weight to the control on the surface This also statistical analysisw. Distributed Parameter State Estimation and Optimal Feedback Control An Experimental Study in Two Space Dimensions GERHARD K LAUSTERER AND W HARMON RAY Abstract Both optimal and suboptimal distributed parameter state estimation om were applied in real time to a heated cylindrical ingot

To address the control problems for nite and in nite dimensional systems the full state information is usually necessary In this thesis an optimal state estimation method is devel oped for spectral distributed parameter systems to account for full state estimation problems with state constraints due to physical limitations In particular a

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