Pneumatic pulse-width modulated pressure control via trajectory optimized fast-switching electromagnetic valves

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1 This ocument contains a post-print version of the paper Pneumatic pulse-with moulate pressure control via trajectory optimize fast-switching electromagnetic valves authore by T. Glück, W. Kemmetmüller, A. Pfeffer, an A. Kugi an publishe in Proceeings of the 13th Mechatronics Forum, International Conference. The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting. Please, scroll own for the article. Cite this article as: T. Glück, W. Kemmetmüller, A. Pfeffer, an A. Kugi, Pneumatic pulse-with moulate pressure control via trajectory optimize fast-switching electromagnetic valves, in Proceeings of the 13th Mechatronics Forum, International Conference, vol. 3/3, Linz, Austria, Sep. 1, pp BibTex Title = {Pneumatic pulse-with moulate pressure control via trajectory optimize fast-switching electromagnetic valves}, Author = {T. Glück an W. Kemmetmüller an A. Pfeffer an A. Kugi}, Booktitle = {Proceeings of the 13th Mechatronics Forum, International Conference}, Year = {1}, Aress = {Linz, Austria}, Month = {Sept }, Pages = {9-99}, Volume = {3/3}, Url = { } Link to original paper: Rea more ACIN papers or get this ocument: Contact: Automation an Control Institute (ACIN) Internet: Vienna University of Technology office@acin.tuwien.ac.at Gusshausstrasse 7-9/E37 Phone: Vienna, Austria Fax:

2 Pneumatic Pulse-With Moulate Pressure Control via Trajectory Optimize Fast-Switching Electromagnetic Valves T. Glück, W. Kemmetmüller, A. Pfeffer, A. Kugi Automation an Control Institute Vienna University of Technology Gusshausstrasse Vienna, Austria glueck@acin.tuwien.ac.at kemmetmueller@acin.tuwien.ac.at pfeffer@acin.tuwien.ac.at kugi@acin.tuwien.ac.at Abstract The esign of a pneumatic pulse-with moulate pressure control via trajectory optimize fast-switching electromagnetic valves is presente. Two fast-switching valves are use in a half brige configuration for pressure control in a pneumatic volume. The valves are operate using a feeforwar control, which guarantees soft laning an time optimality. The control performance an achievable noise reuction of the pulse-with moulate pressure control in combination with the optimize switching strategy is emonstrate by measurement results on an experimental test bench. I. INTRODUCTION In many inustrial applications, there is a eman for pneumatic systems that are controlle by cheap an reliable switching actuators. In automation applications, for instance, pneumatic piston actuators are frequently controlle by means of pneumatic pulse-with moulation, see, e.g., [9], [1], [13], [1]. Here, four fast-switching valves, arrange in a pneumatic full brige, replace the traitional irectional control valve. In orer to achieve a high pulse-with moulation frequency an likewise a high close-loop banwith, short valve switching times are necessary. The high pulse-with moulation frequency, however, prouces acoustic noise an causes mechanical wear of the valves. For this, so-calle soft laning strategies were evelope in recent years, cf. [5], []. In [8], the esign of a feeforwar controller that facilitates soft laning an time optimality of a fast-switching electromagnetic valve was presente. The feeforwar controller was esigne by point-to-point quasi-time-optimal control, which allows to incorporate input constraints in a systematic way. In this contribution, the feeforwar control concept is applie to two fast-switching valves which are arrange in a pneumatic half brige in orer to control the pressure in a chamber by means of pulse-with moulation. For this, the mathematical moel of the chamber is erive an the moel parameters are ientifie by means of measurements. Furthermore, a nonlinear controller base on exact input-output linearization is esigne. At first, in Section II an III, the moeling of the switching valves an the esign of the feeforwar controller are briefly summarize. Subsequently in Section IV the chamber moel is erive an in Section V the pulse-with moulate pressure control is esigne. Measurement results on an experimental test bench, given in Section VI, valiate the performance of the evelope control strategy an emonstrate the resulting noise reuction. Electromagnet II. VALVE MODEL Spring Plunger Working port Exhaust port Sleeve Supply port Sealings Fig. 1: Schematics of the 3/-fast-switching valve. The mathematical moel of the consiere fast-switching valve, schematically epicte in Fig. 1, can be separate into three subsystems: the electromagnetic, the mechanical an the pneumatic subsystem. Since measurement results of the consiere fast-switching valve confirm that the valve is pressure-balance, the pressure forces acting on the plunger will be neglecte. In aition, it is assume that the flow force is small in comparison to the magnetic force. Since no internal feeback from the pneumatic ynamics to the electromechanical subsystem is consiere, the optimal control Post-print version of the article: T. Glück, W. Kemmetmüller, A. Pfeffer, an A. Kugi, Pneumatic pulse-with moulate pressure control via trajectory optimize fast-switching electromagnetic valves, in Proceeings of the 13th Mechatronics Forum, International Conference, vol. 3/3, Linz, Austria, Sep. 1, pp The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

3 problem can be formulate with the pneumatic subsystem being neglecte. A. Electromagnetic Subsystem The equivalent magnetic circuit of the fast-switching valve is shown in Fig. a. It comprises the flux-epenent effective R fc (Φ fc ) Θ Φ fc R l Φ l (a) Reluctance moel. R g (s) R fp Φ fp c v f m, f s, f f v m v Plunger s, w s l s u (b) Schematics. Fig. : Reluctance moel an schematics of the fast-switching valve. core reluctance R fc (Φ fc ), with core flux Φ fc, the effective reluctance R fp of the plunger, the effective reluctance R g (s) of the air gap s between the core an the plunger, an the reluctance R l which accounts for leakage fluxes. The reluctances are moele in the form l fc l fp R fc (Φ fc ) =, R fp =, µ µ fc (Φ fc )A fc µ µ fp A fp R l = l l, R g (s) = s (1). µ A l µ A g Here, l fc, l fp, an l l are the effective lengths of the core, the plunger, an the leakage flux lines, respectively. A fc, A fp, an A l are the corresponing effective areas. The effective length of the air gap is equal to s, since there are two air gaps between the core an the plunger, cf. Fig. 1. The corresponing area is enote by A g. Furthermore, µ enotes the permeability of air an the relative permeability µ fp of the plunger is assume to be constant. Saturation of the core is phenomenologically moele as ( ( ) ( ) ) 1 Φfc Φ fc µ fc (Φ fc ) = k 1 exp k + k 3, () A fc A fc with the constant parameters k j, j = 1,, 3. The equivalent reluctance R of the overall system reas as R(Φ fc, s) = R fc (Φ fc ) + R l (R g (s) + R fp ) R l + R g (s) + R fp. (3) Using the magnetomotive force Θ = Ni of the coil, where i is the current an N is the number of turns, the flux Φ fc through the coil is given in the form Φ fc = Θ R. () Base on the flux linkage ψ = NΦ fc of the coil, Faraay s law yiels t ψ = v Ri, ψ() = ψ (5) with initial conition ψ, the electric resistance R an the applie voltage v. Moreover, the coil current i can be expresse in terms of the flux linkage an the air gap in the form i = B. Mechanical Subsystem R(ψ, s) N ψ. () Fig. b shows a schematic iagram of the forces acting on the plunger of the fast-switching valve. Here, s is the plunger position an w = ṡ is the plunger velocity. The mass of the plunger is enote by m v, the stiffness of the loa spring by c v, an the viscous amping coefficient ue to the friction of the housing an the sealing elements by v. The plunger is loae by the magnetic force f m (ψ, s), the spring force f s (s) an the friction force f f (w). Base on the magnetic energy with (3) an (), see, e.g., [7], W m (ψ, s) = = ψ ψ i( ψ, s) ψ R fc ( ψ/n) N the magnetic force yiels ( ) f m (ψ, s) = s W m (ψ, s) ψ ψ + 1 N R l (R g (s) + R fp ) R l + R g (s) + R fp ψ = 1 R l N (R l + R g (s) + R fp ) The spring force is moele as (7) ( ) s R g (s)ψ. (8) f s (s) = c v (s l c ) (9) with the preloa force c v l c. The friction force is assume to be compose of Coulomb friction an viscous friction, i.e. f f (w) = f fc sign(w) v w, (1) where the Coulomb friction force is enote by f fc an the viscous amping coefficient by v. Henceforth, ( ) the signumfunction in (1) is approximate by tanh w w c sign(w), with w c 1, proviing a continuously ifferentiable friction force f f (w). The balance of momentum for the plunger an Faraay s law (5) with () an the reluctance moel (3) results in the mathematical moel t s = w, s() = s, (11a) t w = 1 ( fm (ψ, s) + f s (s) + f f (w) ), w() = w, (11b) m v R(ψ, s) ψ = v R t N ψ, ψ() = ψ. (11c) Post-print version of the article: T. Glück, W. Kemmetmüller, A. Pfeffer, an A. Kugi, Pneumatic pulse-with moulate pressure control via trajectory optimize fast-switching electromagnetic valves, in Proceeings of the 13th Mechatronics Forum, International Conference, vol. 3/3, Linz, Austria, Sep. 1, pp The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

4 III. TRAJECTORY OPTIMIZATION The mathematical moel (11) with the state vector x = [ s w ψ ] T an with the constraine, affine input u = v U = [ u, u +] can be written in the form t x = f(x) + bu, x() = x, (1) with the initial conition x = [ s w ψ ] T, the vector fiel f an the constant input vector b. The control objective is to fin an optimal control input that guarantees a minimal transition time t f for a setpoint change with (u, x ) (u f, x f ), (13) x() = x, u() = u : = f(x ) + bu, x(t f ) = x f, u(t f )= u f : = f(x f ) + bu f (1) an the terminal conition x(t f ) = x f. Therefore, the inputconstraine point-to-point optimal control problem min J(u) = ϕ(t f ) + u U s.t. tf l(u)t t x = f(x) + bu, x() = x, x(t f ) = x f, u U = [ u, u +] (15) has to be solve. In the quasi-time-optimal case, the terminal cost ϕ(t f ) = t f (1) assures the time optimality an the Lagrange ensity l(u) = 1 ru, (17) with r >, serves as a regularization term in orer to avoi singular arcs. Introucing the Hamiltonian, see, e.g., [], H(x, u, λ) = l(u) + λ T (f(x) + bu), (18) with the ajoint states λ, an applying a time transformation t = t f τ that maps the time interval t (, t f ) onto τ (, 1), the optimal control problem can be reformulate by means of Pontryagin s maximum principle, see, e.g., [3], in form of a two-point bounary value problem, i.e. [ ] [ ] x τ λ = t f(x [ ) f ( x f) T (x ) λ + t b f u ] (19a) u = arg min u U H(x, u, λ ) (19b) with bounary conitions x () = x an x (1) = x f (19c) an the transversality conition H(x, u, λ ) τ =1 = 1 (19) resulting from the free en time t f. Here, the superscript refers to optimal variables. A. Solution of the Quasi-Time-Optimal Control Problem Owing to the input affine system representation (1), the first-orer necessary conition arising from the minimization problem (19b) reas in the unconstraine case as ( ) (x u H, u, λ ) = ru + (λ ) T b = () an can be explicitly solve in the form u = 1 r (λ ) T b. (1) In the constraine case it can be easily seen that the optimal control input takes the form u for u u u = ξ(λ ) = u for u (u, u + ). () u + for u u + Note that in the present case (λ ) T b = λ 3. Consiering the limit case r the optimal control input u switches between the limits u an u + whenever λ 3 changes the sign. Then, the solution of the optimal control problem (15) is a bang-bang control. In aition to (), the secon-orer necessary optimality conition, the so-calle Legenre-Clebsch conition ( ) u H (x, u, λ ), (3) is fulfille, for r >. B. Results of the Trajectory Optimization for Soft-Laning For the opening an for the closing of the valve, respectively, the point-to-point transitions u = v u, x = s u u f = v l, x f = s l ψ u an u = v l, x = s l u f = v u, x f = s u ψ l ψ l ψ u () (5) have to be performe within the normalize transition time τ (, 1). Here, v l, ψ l enote the resulting setpoint voltage an flux linkage at the lower en stop s l an v u, ψ u the setpoint voltage an flux linkage at the upper en stop s u, cf. (1). C. Numerical Results of the Quasi-Time-Optimal Control Problem Numerical solutions of the two-point bounary value problem (19) with (1) an () can be obtaine by utilizing the MATLAB function bvp5c, cf. [11]. The function bvp5c implements a finite ifference metho, in particular a collocation metho, see [11], that controls a scale resiual Post-print version of the article: T. Glück, W. Kemmetmüller, A. Pfeffer, an A. Kugi, Pneumatic pulse-with moulate pressure control via trajectory optimize fast-switching electromagnetic valves, in Proceeings of the 13th Mechatronics Forum, International Conference, vol. 3/3, Linz, Austria, Sep. 1, pp The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

5 an the true error, an aapts the mesh gri. The solver may not fin a solution if the initial guess oes not aequately represent the behavior of the system, see, e.g., [11]. As the aequate initial guess is not easy to fin, especially for the ajoint variables, there is a nee for a systematic solution proceure. The two-point bounary value problem is therefore numerically solve in a sequential proceure that can be summarize as follows: P ϑ, p p amb, ϑ amb p, ϑ ṁ out ṁ in p sup, ϑ sup First, a uniform mesh of M = 3 gri points at the time steps τ k = kt k, k =, 1,..., M, T k = 1/M serve as initial guess for the trajectories x (τ k ), λ (τ k ). A linear interpolation is performe between the bounary conitions (19c) of the states x with the setpoint from () for the opening an (5) for the closing, an zero initial values of the ajoint states λ are assume. Then, the problem is solve recurrently for a sequence of parameters r = r start,..., r en, cf. (17), using the previously receive solution as new initial guess. Figures 3 an show numerical results of the quasitime-optimal point-to-point transition for the opening an for the closing scenario for ecreasing parameters r [1 1, 1, 1 3, 1 ]1/V. Note that for all numerical solutions outsie the vertical ashe lines the initial an final values are hel constant for illustration purposes only. The optimal state trajectories for opening the valve are given in Fig. 3a. Fig. 3c shows the corresponing optimal input voltage v, which converges for smaller parameters r towars a bang-bang control. Fig. shows analogous numerical results for the closing scenario of the valve. Note that in this case the solution of the optimal control problem is not purely bangbang. Whenever λ 3 vanishes, i.e. approximately for the time interval I s = [ ] [ ] τ 1, τ.3,., u vanishes as well. This happens if fm vanishes, implying that only the spring force accelerates the plunger. Parameter stuies have shown that this singular arc can be avoie with a smaller plunger mass m v, a smaller amping coefficient v, a larger spring stiffness c v or a larger preloa force c v l c. IV. CHAMBER AND PNEUMATIC VALVE MODEL In the consiere application, a half brige comprising two fast-switching valves is use for pressure control in a chamber, see Fig. 5. The chamber can be escribe by two inepenent variables, the temperature ϑ an the pressure p, an the governing moel equations may be erive from the conservation of mass an energy, see, e.g., [1]. Stipulating an isentropic process, the change of internal energy U is equal to the sum of the enthalpy flow Ḣ = ṁ inh in ṁ out h out an the applie heat flow Q = Aβ(ϑ amb ϑ), that is t U = ṁ inh in ṁ out h out + Aβ(ϑ amb ϑ), () Fig. 5: Setup with pneumatic half brige. with mass flow ṁ j, specific enthalpy h j, j {in, out}, heat transfer coefficient β, chamber surface A an ambient air temperature ϑ amb. Subsequently, it is assume that the air obeys the ieal gas law pv = R s ϑm, with mass m, pressure p, volume V an specific gas constant R s. In aition, the caloric state equations for the ieal gas u j = c v ϑ j an h j = c p ϑ j, with specific internal energy u j = U j /m, specific enthalpy h j = H j /m an the constant, specific heat capacities c v an c p hol. With R s = c p c v an the constant isentropic exponent κ = c p /c v, the temperature ifferential equation can be irectly inferre form () using the mass balance (9) (κ 1)ϑ ϑ = t pv ( ṁ in (c p ϑ in c v ϑ) ṁ out (c p c v ) ϑ ) + Aβ(ϑ amb ϑ), P (7) with initial temperature ϑ() = ϑ. The change of the gas temperature at the inflow valve can be approximate accoring to an isentropic change ϑ in = ϑ sup ( psup p ) 1 κ κ, (8) where the inex sup refers to the supply variables. The consieration of the mass balance t m = t (ρv ) = ṁ in ṁ out (9) with ρ in (9) ρ = p/(r s ϑ) accoring to the ieal gas law in combination with (7) gives rise to the pressure ifferential equation t p =κr s (ṁin ϑ in ṁ out ϑ ) + κ 1 V V Aβ(ϑ amb ϑ), (3) with initial pressure p() = p. The aiabatic lossless flow through a throttle valve, accoring to ISO [], is given by ϑ ṁ j = C j (s j )p j ρ Ψ(Π j ), j {in, out}, (31) ϑ j Post-print version of the article: T. Glück, W. Kemmetmüller, A. Pfeffer, an A. Kugi, Pneumatic pulse-with moulate pressure control via trajectory optimize fast-switching electromagnetic valves, in Proceeings of the 13th Mechatronics Forum, International Conference, vol. 3/3, Linz, Austria, Sep. 1, pp The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

6 r = /V r = 1 1 1/V r = /V r = 1 1 1/V t f = 1.9 ms t f = 1.37 ms t f = 1.17 ms t f = 1.15 ms s in m w in m/s ψ in Vs normalize time τ λ 1 λ λ normalize time τ v in V i in A f m in N normalize time τ (a) Optimal state trajectories. (b) Optimal ajoint state trajectories. (c) Optimal voltage v, current i, magnetic force f m. Fig. 3: Numerical results of the quasi-time-optimal control problem for the opening motion (s u s l ). r = /V r = 1 1 1/V r = /V r = 1 1 1/V t f = 1.9 ms t f = 1.53 ms t f = 1.39 ms t f = 1.38 ms s in m w in m/s ψ in Vs λ 1 λ λ v in V i in A f m in N normalize time τ normalize time τ normalize time τ (a) Optimal state trajectories. (b) Optimal ajoint state trajectories. (c) Optimal voltage v, current i, magnetic force f m. Fig. : Numerical results of the quasi-time-optimal control problem for the closing motion (s l s u ). Post-print version of the article: T. Glück, W. Kemmetmüller, A. Pfeffer, an A. Kugi, Pneumatic pulse-with moulate pressure control via trajectory optimize fast-switching electromagnetic valves, in Proceeings of the 13th Mechatronics Forum, International Conference, vol. 3/3, Linz, Austria, Sep. 1, pp The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

7 with technical ensity ρ = kg/m 3 an technical temperature ϑ = K. The position-epenent pneumatic conuctances C j are assume to be affine in the valve position s j an rea as C j (s j ) = γ(s u s j ) with constant γ > an upper limit s u for j {in, out}. The flow-through function Ψ(Π j ) in (31) is escribe by [] ( ) Πj Π 1 c Ψ(Π j ) = 1 Π c for Π j > Π c (3) 1 for Π j Π c with pressure ratios Π in = p sup /p, Π out = p/p amb an respective constant, critical pressure ratios Π c >. V. PNEUMATIC PULSE-WIDTH MODULATED PRESSURE CONTROL The usage of a pulse-with moulate valve opening area an likewise conuctance C results in a pulse-with moulation of the mass flow ṁ, which allows to control the chamber pressure. Since the valve ynamics is reasonably C j C max χ j Fig. : Pulse-with moulate conuctance. fast compare to the temperature an pressure ynamics, the instantaneous switching of the valves may be assume in the following. Using a suitable control strategy for the two valves of the half brige as has been iscusse in Section IV, either the maximum conuctance C max or the minimum conuctance C min = of the iniviual valves j {in, out} can be prescribe. The pulse-with moulate conuctances C j (t) for t = k with k Z rea as { Cmax for k < t (k + χ j ) C j (t) =, for (k + χ j ) < t (k + 1) (33) where χ j 1 are the uty ratios an is the fixe moulation perio, cf. Fig.. Obviously, the average values C j = 1 ktpwm (k 1) C j (t) t = C max χ j (3) of the conuctances C j can be irectly etermine by means of the pulse ratios χ j. However, the value of the uty ratios χ j can be solely set at the beginning of each moulation perio. Hence, only the mean value p of the pressure an the mean value of the mass flow ṁ can be controlle via the uty ratios χ j. For the controller esign it is assume that the overall gas t temperature is constant, i.e. ϑ = ϑ in = ϑ = ϑ sup. Then, the ynamics of the mean value p of the pressure p = 1 t t p(τ) τ (35) can be irectly euce from (3) in the form t p = κr ( ) s V ϑ ṁ in ṁ out, (3) with the mass flow mean values ṁ j = 1 t t ṁ j (τ) τ (37) an ṁ j in accorance to (31). The interconnection of the fastswitching valves with ṁ in an ṁ out suggests to introuce the virtual control input ( ) α = ξ ṁ = ξ ṁ in ṁ out, (38) whit the abbreviation ξ = κr s ϑ /V. Thus, the control law α = p η 1 e p η e p,i, (39) with the pressure error e p = p p an a sufficiently smooth esire trajectory p of the mean value of the pressure reners the linear error ynamics ë p + η 1 ė p + η e p =. () The ynamics can be arbitrarily assigne by means of the positive control parameters η 1, η >. The conitional integration t e p,i = e p,c τ e p,c = with if (χ χ max ) (e p > ) if (χ χ min ) (e p < ) else e p (1) is introuce in orer to prevent control winup inuce by the limits χ min χ χ max. Accoring to (38), the mass flows rea as ṁ in = { α ξ for α > for α, ṁ out = { for α α ξ for α <. () Since the supply an ambient pressure, p sup an p amb, are assume to be constant, the mass flow mean value (37) may be written in the form ṁ j = ρ t t C j (s j )Γ(p j ) τ, (3) with the pressure-epenent term Γ(p j ) = p j Ψ(Π j ). Moreover, for small pressure variations p j from the mean values p j the following approximation Γ(p j ) Γ( p j ) + O( p j ), () Post-print version of the article: T. Glück, W. Kemmetmüller, A. Pfeffer, an A. Kugi, Pneumatic pulse-with moulate pressure control via trajectory optimize fast-switching electromagnetic valves, in Proceeings of the 13th Mechatronics Forum, International Conference, vol. 3/3, Linz, Austria, Sep. 1, pp The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

8 with the Lanau-symbol O( ), see, e.g., [15], is utilize. The combination of the this approximation with (3) an (3) finally yiels the uty ratios χ j = 1 ρ Γ( p j )C max ṁ j, j {in, out}, (5) which can be irectly translate into closing times of the respective valves. VI. IMPLEMENTATION After the successful testing of the pulse-with moulate control strategy in several simulations, it was implemente on a test bench, see Figures 5 an 7 for the setup. The test bench ientification task min θ J(θ) = 1 T m s.t. T m ( p(t; θ) pm ) t ) p = f p ( p(t; θ), p() = p, t θ = [ ] T Π c C max, (7) ) with f p ( p(t; θ) accoring to (3), is performe for measurements p m uring the time interval t (, T m ). Just with a little abuse of notation the aitional argument shoul explicitly inicate the epenence on the parameter vector θ. The ientification proceure was performe in MATLAB by means of the function fmincon using the sequential quaratic programming metho in combination with the orinary ifferential equations solver oe15s. Fig. 8 shows ientification results for the charging ( ṁ in > an ṁ out = ) an ischarging ( ṁ in = an ṁ out > ) of the chamber. The results show a goo agreement between measurements an simulations, which imply that the moel approximations are reasonable. Fig. 7: Photograph of the test bench. p in bar p in bar p p m consists of two fast-switching valves an a chamber of volume V =. l. The real-time system DSPACE 15 was use for ata processing with a sampling time of T s = 1 µs an a moulation perio of = 1 ms. Two pressure sensors measure the chamber an the supply pressure, p an p sup, respectively. The latter was controlle by means of a pressurecontrol valve to a constant value of p sup = 7 bar. The pressure mean value (35) is approximate by p(k ) 1 N pwm kn pwm 1 l=(k 1)N pwm p(lt s ), () with N pwm = /T s. The integral part (1) is realize by the Euler-metho, see, e.g., [15]. Experimental results of the trajectory optimization for soft laning were presente in [8]. They show that it is possible to open an close the valve in minimal time with almost zero velocity at the en stops, see [8]. These optimize trajectories are use as feeforwar control within the presente pneumatic pulse-with moulation. A. Parameter Ientification Some parameters of the controller esign moel (3) are unknown, which is why the nonlinear ynamic least-squares 1 3 time in s (a) Charging. 8 1 time in s (b) Discharging. Fig. 8: Ientification results of the pressure mean moel. B. Measurement Results Fig. 9 shows measurement results of the pulse-with moulate control strategy for a sinusoial esire trajectory of the pressure mean value p = p a sin(πft) + p o. In Fig. 9a, the parameters are chosen as p o = 3 bar, p a =.1 bar an f = 5 Hz, whereas in Fig. 9b epicts the results for p o = 5 bar, p a =.1 bar an f = 7 Hz. In both cases, a goo control performance can be achieve with the presente control strategy. C. Noise Reuction To illustrate the noise reuction achieve by the optimize trajectories of the valve, a comparison of the Fast-Fourier transformation of soun recorings from pulse-with moulate pressure control with a simple an with the trajectory optimize valve actuation are epicte in Fig. 1. With the simple switching valve actuation only the maximum voltage to open an the minimum voltage to close is applie to the Post-print version of the article: T. Glück, W. Kemmetmüller, A. Pfeffer, an A. Kugi, Pneumatic pulse-with moulate pressure control via trajectory optimize fast-switching electromagnetic valves, in Proceeings of the 13th Mechatronics Forum, International Conference, vol. 3/3, Linz, Austria, Sep. 1, pp The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

9 p in bar χj p p χ in χ out.1..3 time in s abs ( fft( ) ) /Nfft frequency in khz (a) Simple valve actuation. abs ( fft( ) ) /Nfft frequency in khz (b) Trajectory optimize valve actuation. Fig. 1: Fast-Fourier transformation of soun recorings from a simple an from the trajectory optimize valve actuation for a sinusoial esire trajectory with a frequency of f = 5 khz. p in bar χj (a) Sinusoial reference trajectory with p a =.1 bar, p o = 3 bar, f = 5 Hz χ in p p.1..3 time in s χ out (b) Sinusoial reference trajectory with p a =.1 bar, p o = 5 bar, f = 7 Hz. Fig. 9: Measurement results of the pulse-with moulate pressure control for ifferent sinusoial esire reference trajectory of the pressure mean value p = p a sin(πft) + p o. valves. The results in Fig. 1 reveal a significant reuction in the frequency range Hz. VII. CONCLUSION In this work, pulse-with moulate pressure control with fast-switching valves arrange in a half brige is presente. The fast-switching valves are riven by optimize feeforwar trajectories which facilitate soft laning an time optimality. For this, a point-to-point quasi-time-optimal control problem is formulate by means of Pontryagin s maximum principle an numerically solve by a irect approach. The erivation of a mean value chamber moel an the ientification of its parameters form the starting point for the esign of a nonlinear pressure controller. Measurement results on an experimental test bench show the applicability of the propose pressure control an emonstrate the achievable noise reuction of the pulse-with moulate pressure control in combination with the optimize switching strategy. Future work aresses the extension of the control strategy to pneumatic piston actuators. ACKNOWLEDGMENT The authors woul like to thank Festo AG [1] for the financial support an cooperation uring this joint project. REFERENCES [1] Festo AG, June 1. [] ISO 358, June 1. [3] M. Athans an P. Falb. Optimal control: An introuction to the theory an its applications. McGraw-Hill, New York, 19. [] A.E. Bryson an Y.C. Ho. Applie optimal control. John Wiley & Sons, New York, [5] R. R. Chlany an C. R. Koch. Flatness-base tracking of an electromechanical variable valve timing actuator with isturbance observer feeforwar compensation. IEEE Transactions on Control System Technology, 1():5 3, 8. [] S. K. Chung, C. R. Koch, an A. F. Lynch. Flatness-base feeback control of an automotive solenoi valve. IEEE Transactions on Control System Technology, 15():39 1, 7. [7] A.E. Fitzgeral, C. Kingsley, an S.D.Umans. Electric maschinery. McGraw-Hill, th eition, 3. [8] T. Glück, W. Kemmetmüller, an A. Kugi. Trajectory optimization for soft laning of fast-switching electromagnetic valves. In Proceeings of the 18th IFAC Worl Congress, pages , Milano, Italy, Aug. 8 - Sept [9] S. Hogson, M. Q. Le, M. Tavakoli, an M. T. Pham. Sliing-moe control of nonlinear iscrete-input pneumatic actuators. In Proceeings of the IEEE/RJS International Conference on Intelligent Robots an Systems, pages , San Francisco, USA, -3 Sept. 11. [1] K.A. Hofmann, S.T.L Chiang, M.S. Siiqui, an M. Papaakis. Funamental equations of flui mechanics. Engineering Eucation System, Wichita, 199. [11] J. Kierzenka an L.F. Shampine. A bvp solver that controls resiual an error. Journal of Numerical Analysis, Inustrial an Applie Mathematics, 3(1-):7 1, 8. [1] A. Messinaa, N. I. Giannoccaroa, an A. Gentile. Experimenting an moelling the ynamics of pneumatic actuators controlle by the pulse with moulation technique. Mechatronics, 15: , 5. [13] T. Nguyen, J. Leavitt, F. Jabbari, an J. E. Bobrow. Accurate sliingmoe control of pneumatic systems using low-cost solenoi valves. IEEE/ASME Transactions on Mechatronics, 1():1 19, 7. [1] X. Shen, J. Zhang, E. J. Barth, an M. Golfarb. Nonlinear moel-base control of pulse with moulate pneumatic servo systems. Journal of Dynamic Systems, Measurement, an Control, 18:3 9,. [15] J. Stoer an R. Burlisch. Introuction to numerical analysis. Springer, New York,. Post-print version of the article: T. Glück, W. Kemmetmüller, A. Pfeffer, an A. Kugi, Pneumatic pulse-with moulate pressure control via trajectory optimize fast-switching electromagnetic valves, in Proceeings of the 13th Mechatronics Forum, International Conference, vol. 3/3, Linz, Austria, Sep. 1, pp The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.