A Model Free Automatic Tuning Method for a Restricted Structured Controller. by using Simultaneous Perturbation Stochastic Approximation

Transcription

1 28 Aerican Control Conference Westin Seattle Hotel, Seattle, Washington, USA June -3, 28 WeC9.5 A Model Free Autoatic Tuning Method for a Restricted Structured Controller by Using Siultaneous Perturbation Stochastic Approxiation (SPSA) QingHui Yuan Abstract A odel free auto tuning algorith is developed by using Siultaneous Perturbation Stochastic Approxiation (SPSA). For such a ethod, plant odels are not required. A set of closed loop experients are conducted to generate data for an online optiization procedure. The optiu of the paraeters of the restricted structured controllers will be found via SPSA algorith. Copared to the conventional gradient approxiation ethods, SPSA only needs the sall nuber of easureent of the cost function. It will be beneficial to application with high diensional paraeters. In the paper, a cost function is forulated to directly reflect the control perforances widely used in industry, lie overshoot, settling tie and integral of absolute error. Therefore, the proposed auto tuning ethod will naturally lead to the desired closed loop perforance. A case study of auto tuning of spool position control in a twin spool two stage valve is conducted. Both siulation and experiental study in TI C2 target deonstrate effectiveness of the algorith. I. INTRODUCTION Autoatic tuning of PID controllers have been one of the active research areas for decades. Autoatic tuning aes it possible to autoatically generate gain schedules. The design ethods of autoatic tuning can be classified into three categories () feature based techniques; (2) analytical ethods; and (3) optiization []. The feature based autoatic tuning includes Ziegler-Nichols ethods, and its failies with further odification [2]. In tie doain, Ziegler-Nichols ethods can directly provide the gains based on the doinant dead tie and the doinant tie constant of the step response. In the frequency doain, the ultiate gain and ultiate frequency will directly offer the PID gains through the tuning forula [2]. However, it is well nown that Ziegler-Nichols design criterion gives an underdaped syste perforance, which is undesirable for any cases. When it coes to analytical ethods, we have poleplaceent analysis, Internal Model Controller (IMC), and so on. Analytical ethod can provide the predictable perforance if the dynaics of open loop syste are nown. For optiization based ethods, Modulus Optiu (BO) and Syetrical Optiu (SO) are the ost often used ethods in frequency doain. They attept to obtain the optial loop transfer function, fro which the desired controller is constructed. As aforeentioned, ost of available auto-tuning ethods need plant odels. Nevertheless, plant odels are soeties difficult/expensive to obtain. A first principle based odel usually needs a lot of effort to develop and to validate; while a detailed epirical odel requires a large aount of data fro the process. In addition, variation of dynaics fro Q. Yuan is with Eaton Corp. Innovation Center, Eden Prairie, MN 55344, USA (eail: QinghuiYuan@eaton.co). one coponent to another due to anufacturing tolerance aes it difficult to apply an identical plant odel to different instances. Therefore, an online odel free tuning ethods will be of significant interest in industry. For such a ethod, plant odels will not be required. A set of specific closed loop experients, involving both the actual unnown plant and the proposed controller, are conducted to generate data for use in an online optiization procedure. A cost function will be defined in association with the closed loop perforance. The online optiization will locate the optial values of the corresponding paraeters of the controller. The odel free auto tuning approach is suitable for restricted structure controller whose structure and paraeters have been predefined. Such a controller could be linear, lie phase-lead, phase-lag or PID controllers [3, Chapter 3], or nonlinear. For odel free optiization based auto-tuning algorith, the critical part is the recursive search algorith. In industry, coputation capability of ebedded electronics is usually liited due to cost constraints. However, the fast convergence of the algorith is preferred in order to iprove productivity and reduce downtie. Soe typical optiization algoriths, lie Newton-Raphson and Steepest Descent, will be effective if good gradient inforation is available. In any cases, the gradient is not given and can only be approxiated. For those ethods lie finite difference equations, the larger nuber of gain perturbations and syste response generation are needed. There have been soe efforts of generating gradient with less coputation. For exaple, Controller Paraeter Cycling Tuning [Page. 28] [3] uses trigonoetric function orthogonality to extract gradient and Hessian inforation. The gradient can be obtained via sine gain perturbation - sine extraction, while the Hessian atrix can be calculated via sine gain perturbation - cosine extraction. However, the proble is that the nuber of easureent of the cost function will draatically increases as the diension of paraeters expands. In this paper, we will adopt Siultaneous Perturbation Stochastic Approxiation (SPSA) into auto-tuning algorith. SPSA is a unique optiization ethod that can efficiently approxiate gradient via the cost function easureent. The benefit of SPSA is that the nuber of easureent of the cost function is independent of the diension of the paraeters. In other words, this ethod is especially efficient in high diensional probles. A good solution can be offered by for a sall nuber of easureent of the cost function. The rest of the paper will be organized as follows: In section II, the diagra of auto-tuning algorith is introduced. The details of cost function forulation and the /8/$ AACC. 539

2 θ Auto-tuning Algorith y od ST ( y ( n ), n ) OS( y ( ), n ) θn y d θ OS( y ( ), n) θn y d u C (θ ) P(s) y Trajectory y T ST ( y ( θ ), n) n (s) G cl. Evaluate the cost function 2. Update paraeters to θ n (n-)t L( θ n, n ) nt. Evaluate the cost function 2. Update paraeters to θ n L( θ, n) n Fig.. The diagra of the auto-tuning algorith, in which G cl (s) is the desired closed loop syste transfer function, P(s) is the actual (nonlinear) plant, C(θ) is the paraetric controller where θ is the control paraeters, y is the easured output, and y od is the output fro the reference odel G cl (s). SPSA algorith is presented, and auto tuning procedure is developed. In Section III, the case study of autoatic tuning of spool position controller of an Electro-Hydraulic valve is conducted, both in siulation and experientally. The concluding rears are included in Section IV. II. AUTOMATIC TUNING ALGORITHM FOR A RESTRICTED STRUCTURED CONTROLLER The diagra of the auto-tuning algorith is illustrated in Fig., where P(s) is the plant, C(θ) is a restricted structured controller that consists of the tie invariant paraeters θ R p where p is the diension of the paraeters, and G cl (s) is a reference odel G cl (s) representing a desired transfer function fro the set point to the output. For siplicity, it will usually a low order approxiation. Auto-Tuning Algorith is located in the upper part of the diagra. It continuously taes the outputs fro the actual plants and the reference odels. The search algorith is used to recursively update the paraeters in order to reduce the cost function. The cost function can be selected to reflect the error between the reference odel output and the actual output. The tie doain based ethod is adopted due to its easier use and less coplexity copared to the frequency doain based ethod [4]. Aong available tie based ethods, step response is very effective. Note that the closed loop step response ethod described here is different fro the conventional ones. In [], for exaple, step response of the plant is introduced, syste ID is used to estiate the odel, and then the gains of the controller can be set based on the obtained plant odel. By contrast, what to be addressed is the closed loop step response. In Fig., it is the set point y d, instead of control coand u, that will be step trajectory. The ultiate goal is to find the optial gain θ such that y approaches to y od. Such a ethod ay even be applied to the plant that is open loop unstable. In this section, the cost function of the auto tuning proble is forulated to directly reflect the control perforance specs widely used in industry. Then, the conventional SPSA, as well as its odified version in order to enhance convergence, Fig. 2. Step response profile and Discrete Distributed (DD) cost function. is presented. Finally, SPSA based autoatic tuning procedure is introduced. A. Discrete Distributed (DD) Cost Function In any control syste design, the perforances lie settling tie, overshoot, and integral of the absolute error, are of prie iportance. It would be very useful if auto tuning process can directly help to achieve the perforance goals. A cost function needs to be forulated to capture the associated perforance. These specs are related to the full cycle of the step profile. Therefore, should a cost function be constructed to reflect these specs, it can only be evaluated after all the inforation for a period have been collected. In other words, for a periodic step response, cost functions will be evaluated in the end of each period, as illustrated in Fig. 2. A cost function is defined as (n)t L(θ, n) = w y (θ) y od dt w 2 OS(y (θ), n) OS d w 3 ST(y (θ),n) ST d () in which L(θ, n) is a cost function for a given paraeter θ in the tie span t [nt, (n)t) where n =, 2, and T is the period of the step profile. y is the easureent, y od is the output of the reference odel, OS(x,n) : {R z,r} R is the apping of the trajectory x R z to the overshoot in the tie span t [nt, (n )T), OS d R is the desired overshoot, ST(x,n) : {R z,r} R is the apping of the trajectory x R z to the settling tie in the tie span t [nt,(n )T), and ST d R is the desired settling tie, w i > for i =,2, 3 are the weighting functions for integral of the absolute error, overshoot and setting tie. The cost function in Eq. () is unique due to its discreteness and distributiveness. Firstly, the cost function is evaluated in the end of each step response cycle. Such a process is a discrete event with the frequency of /T. Secondly, given a set of paraeters, one cycle is needed to evaluate a cost function. For ore than one cost function, they have to be distributed and evaluated in ultiple cycles. In Fig. 2, for exaple, two cost functions, L(θ n, ) and L(θ n, ), are evaluated at (n )T and nt, respectively. nt 54

3 The tuning process can be forulated as an optiization proble: θ = arg in θ L(θ, n) (2) where θ is the optial paraeters that iniize the cost function for all n. B. Conventional Siultaneous Perturbation Stochastic Approxiation (SPSA) For such an optiization proble, the search algorith is ey. Note that any popular algoriths, lie Newton- Raphson and Steepest Descent, cannot be used directly since the gradient inforation is unavailable. We have to resort on the gradient approxiation. In [5], the coparative study has been conducted for various gradient approxiation ethods. It has been found that Siultaneous Perturbation Stochastic Approxiation (SPSA) is the preferable algorith than the standard finite difference SA (FDSA) and the rando direction SA (RDSA). The generic recursive SA procedure is [6] ˆθ = ˆθ a ĝ(ˆθ ) (3) where ˆθ R p is an approxiate of the solution θ at th step of recursion, {a } is a sequence of positive scalars tat approaches zero gradually, ĝ( ) R p is an approxiation of the gradient g( ), and =, 2,3, counts the iterations of the algorith. In general, the SPSA gradient approxiation for g(ˆθ ) is ĝ(ˆθ ) = L(ˆθ c, ) L(ˆθ c, ) 2c L(ˆθ c, ) L(ˆθ c, ) 2c 2 L(ˆθ c, ) L(ˆθ c, ) 2c p where {c } is a sequence of positive scalars, R p is a vector with Bernoulli distribution at the th step of iteration where i for i =, 2,,p is the ith coponent of, L(, ) is the cost function (). Note that the the gradient approxiation fro the standard finite difference approxiation is proportional to the nuber of the paraeters. For high diensional paraeters, the evaluation of the objective function will be costly. Since each cost function need one full cycle of step response, tuning process could be tie consuing. By contrast for the conventional SPSA, only two easureents of the cost functions, L(ˆθ c, ) and L(ˆθ c, ), are required [7] [8]. C. Modified SPSA ) Noralization for Bernoulli distribution: In order to balance the convergence in all paraeter diensions, we have to noralize the Bernoulli distribution with respect to the range of paraeters. The probability ass function for i, each eleent of := [, 2,, p ] T, is given by (4).5 x =.5δ( θ i θ i ) f i (x) =.5 x =.5δ( θ i θ i ) otherwise where < δ < is a scalar deterining the update ratio, and θ i and θ i are the lower bound and upper bound of each eleent of θ, respectively, with θ i θ i θ i. 2) Convergence consideration: SPSA has the first order version and the second order version. For the second order SPSA, the siilar technique used to approxiate the gradient can be extended for evaluation of Hessian Matrix. The second order SPSA converges faster, and the first order SPSA will slow down the convergence rate once an optiu is approached. In our application, we would lie to use the first order SPSA aforeentioned due to its siplicity, and find a way to iprove its convergence rate. The following enhanceents has been considered to speed convergence and to increase algorith stability. Iteration rejection for overaggressive update (5) { ˆθ = IR (ˆθ, ˆθ ˆθ if ˆθ ) := ˆθ > M ˆθ otherwise (6) where IR ( ) is a function reflecting iteration rejection for overaggressive update, M > is a large scalar. Iteration rejection based on cost function coparison For the typical SPSA algorith, each iteration needs two evaluation of the cost functions L(ˆθ c, ), L(ˆθ c, ). It has been suggested in [7] that the third evaluation is added on L(ˆθ, ). For the extra cost function evaluation, the benefit would be directly coparison of the cost function between the adjacent cost functions on ˆθ and ˆθ. ˆθ = IR 2 [ˆθ,L(ˆθ, ), L(ˆθ, )] { ˆθ if L(ˆθ :=, ) L(ˆθ, ) > M 2 ˆθ otherwise (7) where IR 2 ( ) is a function reflecting iteration rejection based on cost function coparison, M 2 > is a large scalar. D. Autoatic Tuning Procedure Considering discreteness and distributiveness of the cost function, the overall autoatic tuning procedure is proposed as follows: ) Initialization at t =. = ˆθ =.5(θ θ) ˆθ =.5(θ θ) L O := L(ˆθ, ) = M 3 θ n = ˆθ c (8) where M 3 >, θ n is the paraeter for the next period. In other words, the controller will be in the for of C(θ n ) for the next period. 54

4 θˆ c 3T θˆ c θˆ ˆ θ c 3T ˆ θ c ˆ θ Trajectory L L L o L L o L Tie [sec] Fig. 3. Auto tuning procedure. The values of the paraeters in each step cycle are shown in the top of the profile. Three cycles are needed for ˆθ iteration. 2) At t = (3 2)T, Evaluate the cost function L = L(θ n, 3 2). Update the paraeter for the next cycle evaluation θ n = ˆθ c. 3) At t = (3 )T Evaluate the cost function L = L(θ n, 3 ). Update the paraeter for the next cycle evaluation θ n = ˆθ. 4) At t = 3T a) Evaluate the cost function L O = L(θ n, 3) b) Chec iteration rejections in Eq. (6) (7). Or, evaluate ˆθ = IR (ˆθ, ˆθ ) and ˆθ = IR 2 (ˆθ, L O, LO ). c) Update a, c. d) Generate with probability distribution in Eq. (5). e) The gradient approxiate is rewritten fro Eq. (4) [ ] ĝ(ˆθ ) = L L L T L 2c 2c 2 L L (9) 2c p Note that a variant of gradient approxiation ethod [7] can be easily integrated for soothness. f) Update the paraeter vector ˆθ via ˆθ = ˆθ a ĝ(ˆθ ) () g) Update iteration index =. If > N where N > is a scalar, the procedure stops. h) Chec boundary constraints. For ˆθ i, each eleent of ˆθ, constrain the update within the bound via θ i if ˆθ i < θ i ˆθ i = θ i if ˆθ i > θ i () otherwise for i =, 2,,p. i) Update paraeter for the next cycle evaluation θ n = ˆθ c. j) Go to 2) Since we add the third evaluation of the cost function to iprove convergence rate and stability, 3T is the inial tie required for iteration procedure 2) 4), as illustrated in Fig. 3. ˆθ i Fig. 4. The cross section of Ultronics valve (Courtesy of Eaton Corp.) - ain stage valve bloc, 2 - independent spool for etering, 3 - pilot valve, 4 - voice coil actuator, 5 - centering spring, 6 - pilot spool, 7 - LVDT position sensor, 8 - thin fil pressure sensor, 9 - icrocontroller []. It is worth entioning that the procedure in 4) is ore coputation intensive than ), 2) or 3). However, the tas can be divided and executed in ore than one sapling loop. The error will be negligible since the sapling rate is uch higher than the step response frequency. III. CASE STUDY In this section, an Electrohydraulic syste is targeted to apply the odified SPSA based auto tuning algorith. In this case study, we select a twin spool two stage valve (Ultronics, Eaton Corp., US) as the platfor. The Ultronics valve syste is an advanced electro-hydraulic control valve. It has the unique two-stage twin spool configuration. Unlie its traditional predecessors, the Ultronics valve syste cobines the flexibility of software along with its independent etering spools technology to give the coplete control over the achines hydraulic equipent. Ultronics are used in obile applications in construction, forestry, agriculture, and other arets [9] []. The scheatic of an Ultronics flow control valve is illustrated in Fig. 4. The sensors are ebedded inside the valve so that the valve port pressures and the ain stage spool position can be easured. Therefore, the closed loop control algorith can be loaded into the ebedded electronics to achieve a variety of control objectives, lie pressure/flow/position regulation. Many extra advanced features in the vehicle level application can even be possible. However, ability to control the ain stage spool position is ey for all other functionalities. In this case study, we will consider a restricted structured controller for position regulation. The first principle full order valve odel is highly nonlinear. The detailed description of odeling of such a twin spool two stage valves can be seen in our previous wor []. However, based on the assuption that the pilot stage has faster response copared to the ain stage, we can get the reduced order linear syste approxiation. Siple analysis shows that the control coand is positive associated with the ain stage spool velocity. A velocity feedforward PI controller [2] can be selected for trajectory tracing of spool 542

5 position. u = K d d dt y d K p (y d y ) K i (y d y )dt (2) where y d is the desired position, y is the actual ain stage spool position, u is the current through the voice coil actuator, as can be seen in Fig.. The selected controller falls into the category of a restricted structured controller C(θ) with the paraeters θ = [ K d K p K i ] T. As aforeentioned, for this tie doain based auto tuning algorith, we will use the periodic step response. In the case study, we particularly custoize the step profile during the portion of the steady state range to ephasize tracing (iniizing integral of absolute error). A atheatic representation of such a profile is given by y d = A d O d, t [,.5dT); A d O d A s sin(ω s t φ s ), t [.5dT, dt); O d, t [dt, T) (3) where A d O d are the aplitude and the offset of the step profile, respectively, A s, ω s and φ s are the aplitude, the frequency, and the phase of the sinusoidal signal, d is the duty ratio, T is the period of the step profile, and t := od(t, T). Kp= Ki=3.7 Ki= Kp= Ki=3.7 Ki= Kp= Ki=3.7 Ki= Kp= Ki=3.7 Ki= Kp= Ki=3.7 Ki= Kp= Ki=3.7 Ki= Kp= Ki=3.7 Ki= Kp= Ki=3.7 Ki= A. Siulation study The physical odel of the Ultronics valve is developed in Matlab/Siulin (Mathwors, US). The proposed restricted structured controller in Eq. (2) is ipleented to regulate the plant odel. Monte Carlo siulation is conducted in ters of various cobination of controller gains. As can be seen in Fig. 5 6, we exhaustively visit the possible cobination for K d, K p and K i. Each period of the step profile corresponds to a set of control gains. For instance, the plot in the ost upper left corner of Fig. 5 contains eight step cycles. The first cycle corresponds to a set of gains {K d =.5,K p =.2,K i =.45}, and the second one refers to {K d =.5,K p =.2,K i =.32}, and so on. In Figs. 5 6, K p values increase fro top to botto on the left side, and then fro top to botto on the right side. K i values onotonically increase fro left to right in each plot. Several observations fro Monte Carlo analysis in Figs. 5 6: Larger K i causes greater natural frequency and saller daping ratio, as can be seen in each plot in Fig This is consistent with the closed loop transfer function analysis. As K p increases, the overall perforance is getting iproved. However, if K p is larger than.5, the perforance starts to get degraded draatically. Given K p, the larger K i induces the larger overshoot. On the other hand, it is not clear for the relationship between the overshoot and K p. K d plays a role on deterining optial solution. Copared to the case with K d =.5, the optial solution for K d =.5 shifts fro the larger gains Fig. 5. The ain stage spool trajectory for various control gains. K d =.5. (K p =.4,K i =.7) to the saller ones (K p =.95,K i =.4). This can be explained by the fact that the feedforward ter contributes ore to the control coand as K d increases. The consequence is that the optial proportional and integral gain will be reduced. Next, convexity of the optiization proble is investigated. If this is not a convex proble, then there is no guarantee that the solution will be global optial. Based the collected data in Figs. 5 6, we represent the results in a contour forat in Fig. 7. The figure shows that for each K d, it is a convex proble in the search doain. It ay not preserve its convexity in the entire feasible doain since not all possible solutions are exhaustively investigated. In practice, however, it iplies that we will be solving a convex proble for an appropriately defined doain. The above analysis ensures the convergence of the SPSA auto-tuning algorith. The auto-tuning algorith is ipleented in Stateflow (Mathwors, US). The diagra is illustrated in Fig.. We ω 2 d s 2 2ζ d ω d sω 2 d can define a second order syste G cl (s) = where ζ d and ω d are the desired daping ratio and natural freqency. The auto-tuning process follows the description in Section II-C. Based on the epirical experience, the autotuning doains can be roughly specified as K d [K d, K d ], K p [K p,k p ], and K i [K i,k i ]. The paraeters used in the auto tuning ethod are given as follows: K d =., K d =.75, K p =., K p =.375, 543

6 Kp=.2 Ki=3.7 Ki=.45 Kp=.5 Ki=3.7 Ki= Kp= Ki=3.7 Ki= Kp= Ki=3.7 Ki= Ki Kd = Kp Ki=.45 Kp=.7 Ki= Ki=.45 Kp=.6 Ki= Kd = Kp= Ki=3.7 Ki= Kp= Ki=3.7 Ki= Ki Kp Fig. 6. The ain stage spool trajectory for various control gains. K d is fixed to be.5. K i =., K i =.5, a = ( ).5, c =.25( ).5, M =, M 2 =., M 3 =, δ =, w =.5, w 2 =.3, w 3 =.2. In the auto tuning process, ˆθ is iterated as illustrated in Fig. 8. Note that its iteration rate is 4.5 [sec]. This is consistent with the fact that the iteration period of SPSA tuning will be three ties of that of the step profiles (see Section II-C) with T =.5 [sec]. The convergence rate is relatively fast. After iterations (at t = 45 [sec]), the gains are already very close to the final solution. The optial gains fro the SPSA algorith is {K d =.4,K p =.,K i =.25}. The perforance index (cost function) has been optiized during auto-tuning process. In Fig. 9, the desired and actual ain stage spool positions are illustrated. The left plot shows those in the beginning of auto-tuning; while the right one is the profiles in the end of auto-tuning. The perforance has been significantly iproved. It is worth entioning that instability ay occur during auto tuning process for a certain cobination of paraeter values. In order to protect the syste under test, it will be useful to include the echanis for instability detection and suppression. In the left plot in Fig. 9, vibration occurs at t = 6 [sec]. Instability has been detected at t = 6.5 [sec], and eliinated by resetting the paraeter values. Fig. 7. The contour of the cost function in Eq. with respect to K p and K i. Upper: K d =.5; Lower: K d =.5. Note that the optial solution has changed according to K d. In this case, as K d increases, the feedforward ter contributes ore to the control coand. The consequence is that the optial proportional and integral gain will be reduced. K d K p K i Position [µ ] Fig. 8. y od y ˆθ iteration in the process of auto tuning. Position [µ ] y od y Fig. 9. The desired closed loop syste response y od and the actual ain stage spool position y. Left: the beginning of the auto-tuning; right: the end of the auto-tuning. 544

7 Fig MicroController & Sensors Twin Spool Valve Experiental setup of the twin spool two stage valve. Desired Spool position Actual Spool position Fig.. The tracing perforance of the auto-tuned restricted structured controller for the.5 [Hz] sinusoidal signal. B. Experiental Study The proposed auto-tuning algorith has been validated experientally. The experiental setup is illustrated in Fig.. TI C2 series (TI, US) is selected as target [3]. Target TI C2 Toolbox (Mathwors, US) is integrated with TI code coposer studio for code generation, copiling, lining, and downloading. TI JTAG eulator is used for control and onitoring the signals, shown as a blac bloc in Fig.. In the study, we turn on auto-tuning for two inutes. After the SPSA algorith stops, the tuned optial gains will be autoatically saved. The velocity feedforward PI controller is with the tuned gains K d =.25, K p =.47,K i =.8. Note that the gains fro the experiental study is different fro those fro siulation. The reason is that for the quite coplex plant odel, not all the paraeters are precisely calibrated. Variation of the actual plant fro odel causes different tuning results. In addition, the tuning results for one individual valve are different fro another in experiental study, since anufacturing tolerance will introduce the slightly different dynaic behavior. The level of gain variation depends on the level of variation of electrohydro-echanical property. The position control is tested for the auto-tuned velocity feedforward PI controller. Without loss of generality, a sinusoidal signal with.5 [Hz] frequency and 3 [µ] aplitude is set to be the tracing objective. As shown in Fig., the actual position tracs the desired position quite well for the auto tuned paraeter values of the controller. Note that the tracing perforance is disturbed around y = [µ]. This is caused by nonlinear behaviors of the echanical syste that are not considered by the selected restricted structured controller. IV. CONCLUSION In the paper, a odel free auto tuning algorith for restricted structured controllers is developed. A unique discrete distributed (DD) cost function is defined so that the auto tuning can directly reflect soe typical perforance requireents in industry, lie integral of absolute error, overshoot and settling tie. In addition, the industrial solution needs fast convergence even with the liited coputation capability. In the paper, Siultaneous Perturbation Stochastic Approxiation (SPSA) technology is utilized to optiize the paraeters. The benefit of this approaches is that only three easureent of the cost function are needed independent the diension of the paraeters. It is especially efficient for high diensional proble. The odification of SPSA in ters of Noralization of Bernoulli distribution, and iteration rejection has been ade in order to iprove convergence stability and rate. A case study of auto tuning of a twin spool two stage valve is conducted. The siulation study verifies that the Modified SPSA is effective to locate the optial paraeters of the proposed restricted structured controller. The algorith has also been ipleented in TI C2 ebedded target. The experiental study shows that the auto-tuning process can be successfully executed in the target. The controller with the tuned gains provides a good tracing perforance. REFERENCES [] W. S. Levine, The Control Handboo. Prentice Hall, 996. [2] C. C. Hang, K. J. Astro, and W. K. Ho, Refineents of zieglernichols tuning forula, in IEE, Part D, vol. 38, no. 2, 99. [3] M. A. Johnson and M. H. oradi, PID Control: New Identification and Design Methods. Springer, 25. [4] T. Hagglund and K. J. Astro, Industrial adaptive controllers based on frequency response techniques, Autoatica, vol. 27, no., 99. [5] D. C. 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Yuan and J. Y. Lew, Modeling and control of a two-stage twin-spool valve for energy-saving, in the 25 Aerican Control Conference (ACC), vol. 6, Portland, OR, Jun 25, pp [2] M. Jelali and A. Kroll, Hydraulic Servo-systes: Modeling, Identification and Control. Springer, 23. [3] Texas-Instruents, Ts32 data anual,