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JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1 Visualization of hydraulic cylinder dynamics by a structure preserving nondimensionalization Satoru Sakai, Member, IEEE and Stefano Stramigioli, Fellow, IEEE
Abstract—This paper reveals a new simplicity of a nominal direct search (the brute-force search) approaches [8] in which hydraulic cylinder model whose original representation suffers the computation of the nonlinear dynamics and the updates from too many physical parameters. The 8-dimensional param- of the many physical parameters are repeated in the original eter space in the original representation is reduced to a 3- dimensional parameter space in the proposed nondimensional representation. Also, even if the original parameter space is representation while preserving the parametric structure. To reduced by an usual nondimensional representation, it is never clarify comprehensive relations between the nonlinear dynamics efficient to build and verify a new simulator for the usual and the many physical parameters, an advanced direct search nondimensional representation where the existing simulator for approach is suggested. More precisely, we can repeat the fast the original representation cannot be applied. computation of the nonlinear dynamics and the updates of only three parameters without verifying any new simulator. The On the other hand, such comprehensive relations are already efficient visualization of the numerical solutions presents the best studied for other systems. The mass-damper-spring possible result corresponding to the analytical solution. d2s ds m + d + ks = f Index Terms—Hydraulic systems, nonlinear dynamics, fast dt2 dt computation, visualization. in the original representation is transformed to ∗ ∗ ∗ ∗ ∗ I. INTRODUCTION s¨ + d s˙ + s = f alance between accuracy and simplicity is a key in in a special nondimensional√ representation with only one B modeling for control of hydraulic cylinders. Hydraulic parameter d∗ = d/ mk preserving the parametric structure cylinders are a fluid-mechanical system. However, instead of and the analytical study completely clarified the relations (e.g., the infinite dimensional model, which achieves high accuracy, the critical response at d∗ =2) based on the linearity. The the finite dimensional nominal models are accepted in many Navier-Stokes equations of a fluid system [9] controller design procedures [1] [2] [3] [4]. Nevertheless, such ∂u 1 nominal models are still complex in terms of not only the +(u ·∇)u = − ∇p + ge + νΔu ∂t ρ g nonlinear dynamics (nonlinear response) but also the many physical parameters in the original representation. In fact, in in the original representation is transformed to addition to the well-known mechanical parameters such as the ∂u∗ 1 1 +(u∗ ·∇∗)u∗ = −∇∗p∗ + e∗ + Δ∗u∗ damping constant, the several fluid parameters such as the ∂t∗ Fr 2 g Re bulk modulus and the source pressure can be dominant in the in a special nondimensional representation with only two nonlinear response [5] [6]. Eventually, the comprehensive re- parameters Fr and Re preserving the parametric structure and lations between the nonlinear dynamics and the many physical many numerical studies have clarified the relations. These parameters are not entirely clarified. This implies that even if analytical and numerical studies which are not detailed here a good control result is achieved under a certain experimental provide the foundations for many things today. condition, it may not be justifiable to apply the result to To clarify such relations for a nominal hydraulic cylinder more general conditions. For example, when a linearization model as well, this paper proposes a new special nondimen- based control [7] is useful for a certain experimental hydraulic sional representation that preserves the parametric structure system, it is not clear when the linearization is useful again and suggests an advanced direct search approach different for other hydraulic systems. from the conventional ones with respect to both the efficiency To overcome this situation, due to the nonlinear response, and visualization (3D-visualization), without which many nu- numerical studies are important and relevant. However, since merical studies are less valuable. The proposed nondimen- the original parameter space is too large due to the many phys- sional representation is a new simplicity of the nominal model ical parameters, it is never efficient to apply the conventional that does not exist in more accurate or complex existing Manuscript received September 1, 201x; revised June 2x, 201x. This work models (e.g. [10] [11] in the original representation) as well as was supported by JSPS KAKENHI JP26420173 and Fluid Power Technology in the merely simple (freely-truncated) existing models. More Promotion Foundation. XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX precisely, without building and verifying any new simulator, XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX we can repeat the fast computation of the nonlinear dynamics XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX and the updates of only three parameters in the proposed Satoru Sakai is with Faculty of Mechanical Engineering, Shinshu Uni- versity, Japan. Stefano Stramigioli is with Faculty of Electrical Engineering, nondimensional representation. To provide an example, the University of Twente, Netherlands. numerical existence and nonlinearity are efficiently visualized JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 2
x 10 x 106 since the nonlinear dynamics computations are impossible if 5 4 the numerical existence is not achieved and the linearization + 0 3 is less reliable if the nonlinearity is strong. Input u [m] 2
The rest of this paper is organized as follows. In Section 0 2 4 6 8 10 Pressure p [Pa] 0 2 4 6 8 10 Time [s] Time [s] II, a nominal model of hydraulic cylinders is reviewed in the x 106 6 original representation. A new special nondimensional repre- 500 - sentation is proposed and compared with other nondimensional 0 4 representations in Section III. In Section IV, the effectiveness Force [N] f 2 0 2 4 6 8 10 Pressure p [Pa] 0 2 4 6 8 10 of the proposed nondimensional representation is confirmed. Time [s] Time [s] 0.2 Conclusions are provided in Section V. 0.05 0 0 II. THE NOMINAL MODEL
Velocity [m/s] v 0 2 4 6 8 10 0 2 4 6 8 10 Let us start with the nominal model of Figure 1 in the Time [s] Displacement s [m] Time [s] original representation: [12] [13] [14] [15] ⎧ 2 Fig. 2. Example of the nominal model output (the black curves) ⎪ d s ds (M, D,L,A ,A ,b,C,P)=(14, 3200, 0.075, 7.0 × 10−4, 5.4 × ⎪ M = −D + A+p+ − A−p− with + − ⎪ 2 10−4, 5.3 × 108, 1.6 × 10−4, 7 × 106) ⎪ dt dt and the experimental output [15] ⎨ (the red dots) whose valve is replaced by LSVG-01EH-20-WC-A1-10 (Yuken dp+ b ds − Kogyo). Σ0 ⎪ = A+ + Q+(p+,u) (1) ⎪ dt A+(L/2+s(t)) dt ⎪ ⎪ dp− b ds ⎩ = +A− − Q−(p−,u) where the flow gain C [ m5/kg] and the source pressure P dt A−(L/2 − s(t)) dt [Pa] are the positive constants. The nominal model introduces where the displacement s(t) [m], the cap pressure p +(t) [Pa], the restricted domain s ∈ (−L/2,L/2) and p± ∈ [0,P] and the rod pressure p−(t) [Pa], and the spool displacement (the the absolute notation within the square root functions (2) is input) u(t) [m] are the functions of time t [s]. The subscript + dropped. − and denote the cap-side and the rod-side, respectively, and Remark 1 (Relating to uncertainty) The equations (1) ignore ± the subscript denotes both sides. The driving force is f(t)= the nonlinear friction effect and also the internal and external − A+p+(t) A−p−(t) [N]. The mass M [kg], the damping leakage effects at least. The equations (2) assume the steady ≥ 2 constant D [Ns/m], the piston areas A+ A− [m ], and the flow and the negligible servo dynamics of the zero-lapped bulk modulus b [Pa] are the positive constants. The cylinder spool valve. On the other hand, the stroke L can include − volumes V+(s(t)) := A+(L/2+s(t)), V−(s(t)) := A−(L/2 the pipeline length effect and the bulk modulus b includes s(t)) [m3] with the constant stroke L [m] are the functions 3 the pipeline (or tube) flexibility effect. Figure 2 shows an of the displacement s(t). The input flows Q+ and Q− [m /s], example of the accuracy between the nominal model (1) are approximated by Bernoulli’s principle: (2) and an experimental setup (a real system) in a practical Q+ = B(p+, +u)u, Q− = B(p−, −u)u (2) frequency band (see [15] for details). This figure displays a long time cross validation in which the experimental outputs with ⎧ √ (the red dots) were never used in the parameter identification − ⎨ C r + P (u>0) procedure for the nominal model outputs (the black curves). B(r, u)=⎩ 0(√ u =0) Nevertheless, with respect to the nonlinear responses in the C +r − 0(u<0) pressures and displacement, the nominal model has an accu- racy that any linearized model (transfer function) cannot have. Of course, the difference (e.g. the nonlinear friction effect) Servo between the nominal model and the experimental setup exists Valve V0+ V0 and depends on each experimental setup but would change Pipe Cylinder continuously. In the context of robust control [16] [17], the p difference is uncertainty taken into account in the controller + C design procedure. Db L/2 The nominal model (1) (2) is not our result. Not the u P A+ accuracy but a new simplicity is our contribution evaluated M in terms of the efficiency and visualization. A s L/2 C p III. A SPECIAL NONDIMENSIONALIZATION 0 First, a new special nondimensional representation is pro- posed. Second, the advantages of the proposed nondimensional Spool Piston representation are discussed in comparison with other nondi- mensional ones because they are not unique [18] [19]. Proposition 1 (Special nondimensional representation) Fig. 1. Nominal hydraulic cylinder model. Consider the original representation (1) (2) of the nom- JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 3 inal model. Then, there exists a set of a time scal- and the input transformation u∗ =(1/U)u with U = ∗ ∗ ∗ ∗ T ing t =(1/T )t, a variable scaling (s ,p+,p−) = ( A3 L/M)/C =: U T + s. Then the original form (5) is trans- ((1/S)s, (1/P+)p+, (1/P−)p−) , and an input scaling u∗ = formed to the nondimensional form: (1/U)u by which the original representation (1) (2) is trans- ⎧ ⎡ ⎤ ⎡ ⎤ ⎪ ⎪ 0+100 0 formed to the following nondimensional representation: ⎪ ⎢ − − ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎧ ⎪ ∗ ⎢ 1 D J23 J24 ⎥ ∗ ⎢ 0 ⎥ ∗ ∗ ∗ ∗ ∗ ∗ ⎨⎪ x˙ = ⎣ ∗ ⎦∇x∗ H +⎣ ∗ ∗ ⎦ ⎪ s¨ = −D s˙ + p+ − A p− 0 −J23 00 +J23Q+ ⎪ ∗ ∗ ∗ ∗ ⎪ 0 −J 00 −(J /A )Q ⎨ ⎪ 24 24 − ∗ 1 − ∗ ∗ ⎪ p˙+ = ∗ s˙ + Q+ ⎪ −1 −T −1 ∗ ∗ ∗ Σs 1/2+s (3) ⎪ TsXs FXs TsXs gUu =:g u ⎪ ⎩ ∗ ∗T ∗ ⎪ y := g ∇ ∗ H ⎩⎪ ∗ 1 ∗ 1 ∗ x p˙− = +˙s − Q− (6) 1/2 − s∗ A∗ with and ∗ ∗ ∗ ∗ J23 =+1/(1/2+s ),J24 = −1/(1/2 − s ), Q∗ = B∗(p∗ , +u∗)u∗,Q∗ = B∗(p∗ , −u∗)u∗ (4) + + − − and the nondimensional energy with H∗ =(1/2)(v∗)2 ⎧ √ ∗ ∗ ⎨ −r + P ∗ (u>0) −(1/2+s )(+1 + p+) ∗ ∗ ∗ ∗ B (r, u)= 0(u =0) −(1/2 − s )(+1 + p−)A ⎩ √ +r − 0(u<0) ∗ ∗ ∗ in which D := D L/(MbA+), A := A−/A+, P := P/b. ∗ ∗ ∗ where T , S, P+, P−, and U are the constants and D , A , Again, by the gradient of the nondimensional energy H in and P ∗ are the nondimensional parameters. The notation •˙ the input-state equation of the nondimensional form (6), the denotes the derivative with respect to the nondimensional time nondimensional representation (3) (4) is obtained. t∗. In general, a model is valuable when the model has desirable In the following proof of Proposition 1, the original rep- properties that the other models do not have. Not only accuracy resentation (1) (2) is converted into an input-state equation but also simplicity for control are among the properties. In of a physical form from which the special nondimensional a word, this paper highlights that the nominal model has representation (3) (4) is derived via a set of the state and input high simplicity that more accurate or complex existing models transformation [20] and also the time transformation. (e.g., [10][11][21]) do not have as well as the merely simple Proof of Proposition 1. The original representation (1) (2) is (freely-truncated) existing models do not. A simplicity for converted into an input-state equation of the form [13]: control of the nominal model is the form (5) which is not our ⎧ ⎡ ⎤ ⎡ ⎤ contribution and an application [13] of the physical form [22] ⎪ 0+100 0 [23] developed for the finite dimensional version of physical ⎪ ⎢ − − ⎥ ⎢ ⎥ ⎪ dx ⎢ 1 DJ23 J24 ⎥ ⎢ 0 ⎥ systems. The physical form provides so many links to fruitful ⎨ = ⎣ ⎦∇xH +⎣ −1 ⎦ dt 0 −J23 00 +bV+ Q+ −1 results in modeling and control (e.g., modeling of the infinite ⎪ 0 −J24 00 −bV Q− ⎪ − dimensional systems, robust stabilization, learning) [24] [25] ⎪ ⎩⎪ F gu [26] than the general nonlinear forms (e.g., x˙ = f(x)+g(x)u T y = g ∇xH and y = h(x) [27]). Indeed, the physical form is a special case (5) of the general nonlinear forms. Regarding our contribution, the T with the state x =(s, pm,p+,p−) , rest of this paper reveals that the nominal model has another −1 −1 simplicity for the parametric structure linked to the several J23(s)=+bV+(s) A+,J24(s)=−bV−(s) A−, advantages, that is, the efficiency and visualization. and the original energy Technically speaking, the proposed nondimensional repre- sentation (3) (4) is different from the conventional ones with 2 − − H = pm/(2M) V+(s)(b + p+) V−(s)(b + p−). respect to the visualization (3D-visualization) at a minimum. The time scaling in the famous nondimensionalizations [21] Here, the notation ∇ denotes the gradient with respect to the x is not coupled with the state and input scaling to reduce the variable x. The variable p = Mv is the momentum imparted m parameters for the visualization. Unlike the translational joint by the velocity v = ds . By the gradient of the original energy dt corresponding to the nominal model, the rotational joint [28] H in the input-state equation of the form (5), the equations is complex due to more parameters making the visualization (1) are obtained by a direct calculation. impossible. Also, the translational joint formulation cannot be Since the state x is defined, let us take the set of the time ∗ a special case of the rotational joint formulation [29]. transformation t =(1/T )t with T = (ML)/(bA+)=:Ts, ∗ ∗ ∗ ∗ ∗ T −1 The first advantage of the proposed nondimensional the state transformation x =(s ,v ,p+,p−) = Xs x with ⎡ ⎤ ⎡ ⎤ representation (3) (4) is the parameter space reduc- S 000 L 000 tion. The 8-dimensional parameter space with θ := ⎢ ⎥ ⎢ ⎥ ∈ R8 ⎢ 0 MS/T 00⎥ ⎢ 0 MLbA+ 00⎥ (M,D,L,A+,A−,b,C,P) + in the original representa- Xs := ⎣ ⎦ = ⎣ ⎦ , 00P+ 0 00b 0 tion (1) (2) is reduced to the 3-dimensional parameter space ∗ ∗ ∗ ∗ ∈ R × × R ⊂ R3 ⊂ R8 00 0P− 000b with θ := (D ,A ,P ) + (0, 1] + + + JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 4 in the proposed nondimensional representation (3) (4). Of and the parametric structure differs from that of the original course, other nondimensional representations bring a similar energy H since A∗ ≡ 1. advantage [18] [19]. To discuss the additional advantages of Eventually, the above significant difference is trivially the proposed nondimensional representation (3) (4), let us rephrased as the following time response property of the make our examples of other nondimensional representations. hydraulic cylinders. Let the notation φ[θ, x(0),u(t)] denote Example 1 For the original physical form (5), let us take the state x(t) in the original representation (1) (2) of θ = ∗ the set of the time transformation t =(1/T )t with T = (M,D,L,A+,A−,b,C,P) at time t starting from the initial ∗ (ML)/(bA+)=:T1 and the state transformation x = state x(0) in the presence of the input signal u(τ)(0≤ τ ≤ t). ∗ ∗ ∗ ∗ T −1 (s ,v ,p+,p−) = X1 x with Theorem 1 (Structure preserving property) Suppose the ⎡ ⎤ ⎡ ⎤ state x(t)=φ[θ, x(0),u(t)] exists. Then the nondimensional S 000 L 000 ∗ ∗ −1 ∗ ∗ ∗ ⎢ ⎥ ⎢ ⎥ state x (t )=X φ[θ, Xsx (0),Usu (Tst )] in the special ⎢ 0 MS/T 00⎥ ⎢ 0 MLbA+ 00⎥ s X1 :=⎣ ⎦=⎣ ⎦ nondimensional representation (3) (4) at nondimensional time 00P+ 0 00b 0 ∗ t =(1/Ts)t starting from the nondimensional initial state 00 0P− 000bA+/A− ∗ −1 x (0) = Xs x(0) in the presence of the nondimensional input ∗ ∗ ∗ and the input transformation u∗ =(1/U)u with U = u (t )=(1/Us)u(t ) is given as 3 ( A+L/M)/C =: U1. Then, via a procedure similar to the x∗(t∗)=φ[θ∗ (θ),x∗(0),u∗(t∗)] one in the proof of Proposition 1, we obtain one of the other special (9) nondimensional representations: ∗ ⎧ of θspecial(θ)=(1,D L/(MbA+), 1, 1,A −/A+, 1, 1,P/b ). s¨∗ = −D∗s˙∗ + p∗ − p∗ ⎪ + − 0 ∗ 0 ∗≤1 0 ∗ ⎪
T one in the proof of Proposition 1, we obtain one of the other test initial state x(0) = (0, 0,P/2, (A+/A−)P/2) ∈ ΩLP nondimensional representations: leaves the region ΩLP for the first time in the presence of ⎧ ⎪ ∗ − ∗ ∗ ∗ ∗ − ∗ ∗ a test signal u(t)=Au sin(2πfut) in the test period [0,Tu]. ⎪ s¨ = D s˙ + A+p+ A−p− ⎪ The numerical existence is achieved only if the escape time ⎨⎪ ∗ ∈ ∗ b ∗ 1 ∗ te [0,Tu] does not exist. The numerical existence depends p˙+ = −s˙ + Q+ Σ2 ∗ ∗ (10) on the setting of the test parameters Au, fu, and Tu as well ⎪ 1/2+s A+ ⎪ ∗ as θ =(M,D,L,A+,A−,b,C,P) and x(0). ⎪ ∗ b ∗ 1 ∗ ⎩ p˙− = +˙s − Q− The nonlinearity is evaluated by a difference between an 1/2 − s∗ A∗ − output of the nominal model and that of the linearized model: and ⎧ ⎪ 2 ⎪ d sˆ − dsˆ − ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎪ M 2 = D + A+pˆ+ A−pˆ− Q+ = B (p+, +u )u ,Q− = B (p−, −u )u (11) ⎪ dt dt ⎨⎪ with ˆ dpˆ+ −1 − dsˆ ⎧ √ Σ0 ⎪ = bV+(0) A+ + Q+(P/2,u) (12) − ∗ ⎪ dt dt ⎨ r +1 (u > 0) ⎪ ∗ ∗ ∗ ⎪ dpˆ− −1 dsˆ B (r, u ):=⎩ √0(u =0) ⎩ = bV−(0) +A− − Q−(P/2,u) +r − 0(u∗ < 0) dt dt √ T ∗ ∗ 2 ∗ 2 whose state xˆ(t):=(ˆs(t), sˆ˙(t), pˆ+(t), pˆ−(t)) starts from the in which D := D/ MPL, A+ := A+/L , A− := A−/L , b∗ := b/P same state xˆ(0) = x(0) in the presence of the same input Now, the other nondimensionalization (10) (11) also pre- u(t). The displacement s(t) and the driving force f(t)= − serves the parameter structure in the original representation (1) A+p+(t) A−p−(t) are relevant in control [1] [7] whereas (2). Indeed, by dropping the superscript • ∗, the other nondi- the pressures p±(t) are used in parameter identification [15]. mensional representation (10) (11) can be equal to an original Here, the difference is defined as the FIT ratio [32]: ⎛ ⎞ representation (1) (2) when θ =(1,D,1,A+,A−,b,1, 1). The Te nondimensional energy of the other nondimensional represen- ⎜ i i 2 ⎟ ⎜ (ˆy0(t) − y0(t)) ⎟ tation (10) (11) described by ⎜ =0 ⎟ i ⎜ − t ⎟ × ∗ ∗ 2 FIT(y0)=⎜1 ⎟ 100 H =(1/2)(v ) ⎜ Te ⎟ − ∗ ∗ ∗ ∗ ⎝ i i 2 ⎠ (1/2+s )(b + p+)A+ (y0(t) − y¯0) ∗ ∗ ∗ ∗ −(1/2 − s )(b + p−)A−, t=0
i i can be a special case of the original energy H. where y¯0 is the mean value of the i-th element y0(t) (i = T The fourth advantage of the proposed nondimensional rep- 1, ··· , 4) of the outputs y0(t):=(p+(t),p−(t),f(t),s(t)) i resentation is the visualization. A difference between the of the nominal model and yˆ0 is the i-th element of the T nondimensionalizations in Proposition 1 and Example 2 is corresponding outputs yˆ0(t):=(ˆp+(t), pˆ−(t), fˆ(t), sˆ(t)) of the dimension of the parameter space. Of course, the 4- the linearized model (12). The results on the velocity v(t) can ∗ ∗ ∗ ∗ ∈ R4 dimensional parameter space with (D ,A+,A−,b ) + be discussed by that on the displacement s(t). If the numerical is much smaller than the original 8-dimensional parameter ∈ 8 existence is achieved, Te := Tu, otherwise Te := te [0,Tu]. R i space + and close to the proposed 3-dimensional parameter The value of FIT(y0) can be negative. space with θ∗ =(D∗,A∗,P∗). However, only the proposed 3- dimensional parameter space can be visualized in 3D whereas even the 4-dimensional parameter space cannot. B. Experimental conditions The nonlinear dynamics computation and the parameter IV. 3D VISUALIZATION updates were repeated in the proposed nondimensional rep- Not the accuracy but the new simplicity is evaluated in terms resentation instead of the original representation. For the nonlinear dynamics computations, the equation (9) was applied of the efficiency (parameter space reduction, verification-free, ∗ ∗ and fast computation) and visualization. to compute the nondimensional state x (t ) starting from the initial state x∗(0) = (0, 0,P∗/2,A∗P ∗/2)T in the presence ∗ ∗ ∗ ∗ of the test signal Au sin(2πfu t ) with the amplitude Au := A. The numerical existence and nonlinearity ∗ ∈ Au/Us =0.01 and the frequency fu := Tsfu [0.001, 10]. ∗ ∗ For many practical nonlinear systems, one of the most The test period was defined as [0,Tu ]:=[0, 5/fu]. The fundamental properties may be the stability [30] as long as the modified backward differential formula with the variable step state exists. Especially, it is relevant that the state x(t) exists was applied (20-sim, Ver. 4.1, 64-bit 2.60GHz CPU with 8GB 2 ∗ ∗ within an restricted region: ΩLP =(−L/2,L/2)×R×[0,P] of memory). The nondimensional outputs y 0 (t ) were given in the presence of the input. In addition to the numerical by the nondimensional state x∗(t∗) directly. Using a similar ∗ ∗ existence, the nonlinearity (or the input-output linearity) is also procedure, the corresponding estimated outputs yˆ0 (t ) were of interest for the linearization based controls [31] [7]. also given by linearized model (12) in the nondimensional The numerical existence is evaluated by the existence of the version. The damping constant, the rod area, and the source ∗ ∗ ∗ ∗ ∗ ∗ ∗ escape time te [30] at which the state x(te) starting from a pressure (D ,A ,P ) ∈ [Dmin,Dmax] × [Amin,Amax] × JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 6
∗ ∗ −5 ∗ ∗ [Pmin,Pmax]=[0.0006, 11.2] × [0.5, 1.0] × [1.4 × 10 , 0.07] to Figure 4 the estimated outputs yˆ0 (t ) are depicted as ∗ ∗ ∗ were updated with the increments δD =1.12, δA =0.05, the curves. For the pressures p±(t), remarkably, around a ∗ and δP ∗ =0.007, respectively and the other parameters frequency fu =0.02, the lower nonlinearity (higher linearity) ∗ ∗ ∗ (M,A+,L,b,C)=(1, 1, 1, 1, 1) were not updated. was achieved uniformly in D A P space. This was observed as the time-response examples in Figure 4(d) and Figure 5(d). ∗ ∗ C. Experimental results and discussion At other frequencies, the pressures p±(t ) and the estimated ones pˆ∗ (t∗) could be very different as shown in Figure Figure 3 shows the numerical existence and nonlinearity ± ∗ ∗ ∗ 4(b) and Figure 5(b) in spite of the best initial condition visualized in D A P space. In the colorless regions on the p∗ (0) =p ˆ∗ (0). In Figure 2, these nonlinearities, the non- slices, the numerical existence is not achieved since the escape ± ± ∗ ∗ ∗ ∗ negative and multi-peak pressures in CASE-c of Figure 4(b), time t ∈ [0,T ] exists such that x (t ) ∈ Ω1 ∗ . Figure 4 e u e P were already observed. shows the time response examples corresponding to Figure 3 ∗ ∗ ∗ ∗ ∗ ∗ ∗ For the driving force f (t )=p+(t ) − A−p−(t ), the in the following four cases: ∗ ∗ ∗ ∗ lower nonlinearity was achieved at every frequency f u in CASE-a: (D ,A ,P )=(0.0006, 0.5, 0.07), ∗ ∗ ∗ ∗ ∗ ∗ D A P space uniformly. Figure 4(c) and Figure 5(c) show CASE-b: (D ,A ,P )=(0.0006, 0.9, 0.07), ∗ ∗ ∗ the corresponding time response examples. Interestingly, even CASE-c: (D ,A ,P )=(6.0, 0.5, 0.07), ∗ ∗ ∗ ∗ ∗ ∗ ∗ when the pressures p±(t ) and the estimated ones pˆ±(t ) were CASE-d: (D ,A ,P )=(6.0, 0.9, 0.07). ∗ ∗ ∗ ∗ very different, the force f (t ) could be approximated roughly The outputs y0 (t ) are depicted as the curves. The maximum ˆ∗ ∗ ∗ 4 by the estimated one f (t ). At every high frequency 10
p* 100 p* 100 p* 100 p* 100 Pressure + Pressure - Pressure + Pressure - 0.08 50 0.08 50 0.08 50 0.08 50 * * 0.06 0 0.06 0 * 0.06 0 * 0.06 0 P P P P 0.04 0.04 0.04 0.04 -50 -50 -50 -50 0.02 0.02 0.02 0.02 -100 -100 -100 -100 0 0 0 0 10 -150 10 -150 10 -150 10 -150 1 1 1 1 5 5 5 5 -200 -200 * 0.5 * 0.5 * 0.5 -200 * 0.5 -200 D 0 * D 0 * D 0 D 0 A A A * A *
100 100 * Force f* Displacement s* Force f* 100 Displacement s 100 0.08 50 0.08 50 0.08 50 0.08 50 * * 0.06 0 0.06 0 * 0.06 0 * 0.06 0 P P P P 0.04 0.04 0.04 0.04 -50 -50 -50 -50 0.02 0.02 0.02 0.02 -100 -100 -100 -100 0 0 0 0 10 -150 10 -150 10 -150 10 -150 1 1 1 1 5 5 [1] 5 5 * -200 * -200 -200 -200 D 0 0.5 D 0 0.5 D* 0 0.5 D* 0 0.5 A * A * * * * A * A (a) f u = 0.001 (e) f u = 0.1 100 100 100 100 p* p* p* p* Pressure + Pressure - Pressure + Pressure - 0.08 50 0.08 50 0.08 50 0.08 50 * * * 0.06 0 * 0.06 0 0.06 0 0.06 0 P P P P 0.04 0.04 0.04 0.04 -50 -50 -50 -50 0.02 0.02 0.02 0.02 -100 -100 -100 -100 0 0 0 0 10 -150 10 -150 10 -150 10 -150 1 1 1 1 5 5 5 5 -200 -200 -200 -200 D* 0 0.5 D* 0 0.5 D* 0 0.5 D* 0 0.5 A * A * A * A *
100 100 100 * 100 Force f* Displacement s* Force f* Displacement s 0.08 50 0.08 50 0.08 50 0.08 50 * * * 0.06 0 * 0.06 0 0.06 0 0.06 0 P P P P 0.04 0.04 0.04 0.04 -50 -50 -50 -50 0.02 0.02 0.02 0.02 -100 -100 -100 -100 0 0 0 0 10 -150 10 -150 10 -150 10 -150 1 1 1 1 5 5 5 5 -200 -200 -200 -200 D* 0 0.5 D* 0 0.5 D* 0 0.5 D* 0 0.5 A * A * A * A * * * (b) f u = 0.002 (f) f u = 0.3
p* 100 p* 100 p* 100 p* 100 Pressure + Pressure - Pressure + Pressure - 0.08 50 0.08 50 0.08 50 0.08 50 * * 0.06 0 0.06 0 * 0.06 0 * 0.06 0 P P P P 0.04 0.04 0.04 0.04 -50 -50 -50 -50 0.02 0.02 0.02 0.02 -100 -100 -100 -100 0 0 0 0 10 -150 10 -150 10 -150 10 -150 1 1 1 1 5 5 5 5 * 0.5 -200 * 0.5 -200 * 0.5 -200 * 0.5 -200 D 0 * D 0 * D 0 D 0 A A A * A *
* Force f* 100 Displacement s* 100 Force f* 100 Displacement s 100 0.08 50 0.08 50 0.08 50 0.08 50 * * 0.06 0 0.06 0 * 0.06 0 * 0.06 0 P P P P 0.04 0.04 0.04 0.04 -50 -50 -50 -50 0.02 0.02 0.02 0.02 -100 -100 -100 -100 0 0 0 0 10 -150 10 -150 10 -150 10 -150 1 1 1 1 5 5 5 5 * 0.5 -200 * 0.5 -200 * 0.5 -200 * 0.5 -200 D 0 * D 0 * D 0 * D 0 * A A A * A (c) f u = 0.003 (g) f u = 0.5
100 100 100 100 p* p* p* p* Pressure + Pressure - Pressure + Pressure 0.08 50 0.08 50 0.08 50 0.08 - 50
* 0.06 0 * 0.06 0 * 0.06 0 * 0.06 0 P P P P 0.04 0.04 0.04 0.04 -50 -50 -50 -50 0.02 0.02 0.02 0.02 -100 -100 -100 -100 0 0 0 0 10 -150 10 -150 10 -150 10 -150 1 1 1 1 5 5 5 5 -200 -200 -200 -200 D* 0 0.5 D* 0 0.5 D* 0 0.5 D* 0 0.5 A * A * A * A *
100 100 100 * 100 Force f* Displacement s* Force f* Displacement s 0.08 50 0.08 50 0.08 50 0.08 50
* 0.06 0 * 0.06 0 * 0.06 0 * 0.06 0 P P P P 0.04 0.04 0.04 0.04 -50 -50 -50 -50 0.02 0.02 0.02 0.02 -100 -100 -100 -100 0 0 0 0 10 -150 10 -150 10 -150 10 -150 1 1 1 1 5 5 5 5 -200 -200 -200 -200 D* 0 0.5 D* 0 0.5 D* 0 0.5 D* 0 0.5 A * A * A * A * * * (d) f u = 0.02 (h) f u = 10
Fig. 3. Numerical existence and nonlinearity in D∗A∗P ∗ space JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 8
0.01 0.01 0.06 p* 0.06 c p* + a +
[1] * 0.005 [1] * 0.005 p d p * ± * ± [1] [1] - b - * * p p 0.04 d 0.04 b u u 0 b 0 b d d a c c Input Input 0.02 0.02 -0.005 a -0.005 Pressure Pressure c a -0.01 0 -0.01 0 0 2000 4000 0 2000 4000 02040 20 40 Time t* [1] Time t* [1] Time t* [1] Time t* [1]
c = (6.0,0.5,0.07) 0.5 c 0.5 0.01 0.01 [1] d = (6.0,0.9,0.07) [1] * * a
d s s b 0.005
0.005 [1] a [1] a c
c * * a = (0.0006,0.5,0.07) b f f d 0 b = (0.0006,0.9,0.07) 0 0 0 b d -0.005 -0.005 Force Force -0.01 -0.01 Displacement Displacement -0.5 -0.5 0 2000 4000 0 2000 4000 02040 0400 Time t* [1] Time t* [1] Time t* [1] Time t* [1] * * (a) f u = 0.001 (e) f u = 0.1
a 0.01 0.01 * 0.06 a p c p* + 0.06 +
0.005 [1] b * b p 0.005 [1] p* * ± [1] - * ±
[1] -
* d *
p d 0.04 d p 0.04 d u 0 u 0 c b c b
Input 0.02 -0.005 c Input -0.005 0.02 a Pressure a Pressure -0.01 0 -0.01 0 0 1000 2000 0 1000 2000 0 5 10 15 0 5 10 15 * * Time t [1] Time t [1] Time t* [1] Time t* [1] a c c 0.5 b 0.5 0.01 [1] 0.01 d d [1] * * s 0.005 s [1] 0.005 [1] *
* a c f 0 a b 0 f 0 0 c b d d -0.005 -0.005 a Force Force -0.01 -0.01 b Displacement Displacement -0.5 -0.5 0 1000 2000 0 1000 2000 0 5 10 15 0 5 10 15 * * Time t [1] Time t [1] Time t* [1] Time t* [1] * * (b) f u = 0.002 (f) f u = 0.3
0.01 c 0.01 a c a p* p* 0.06 + 0.06 +
[1] * 0.005 d 0.005 [1] * p b p * ± * ± [1] - [1] - * * b p 0.04 b p 0.04 d d u 0 b u 0 d c a c Input -0.005 0.02 Input -0.005 0.02 Pressure a Pressure -0.01 0 -0.01 0 0 500 1000 1500 0 500 1000 1500 05100510 Time t* [1] Time t* [1] Time t* [1] Time t* [1]
0.5 a c 0.5 c 0.01 b 0.01 [1] [1] * * s 0.005 d s
[1] 0.005 [1] * b * c a c f 0 0 f 0 0 a d b d -0.005 -0.005 b a Force Force
-0.01 d -0.01 Displacement Displacement -0.5 -0.5 0 500 1000 1500 0 500 1000 1500 05100510 Time t* [1] Time t* [1] Time t* [1] Time t* [1] * * (c) f u = 0.003 (g) f u = 0.5
0.01 0.01 a c a 0.06 p* 0.06 p* c + + [1] 0.005 [1] p* 0.005 p*
b * ± [1] * ±
[1] - -
* b d * p p 0.04 0.04 u
u d d 0 0 a b c d b c
Input 0.02 Input -0.005 0.02 a -0.005 Pressure Pressure
-0.01 0 -0.01 0 0 100 200 0 100 200 0 0.2 0.4 0 0.2 0.4 * * Time t* [1] Time t* [1] Time t [1] Time t [1]
c 0.5 0.5
0.01 0.01 [1] [1] * d * s s 0.005 0.005 [1] [1] a c c
* a * b f f 0 0 0 a 0 b d b d a b c d -0.005 -0.005 Force Force -0.01 -0.01 Displacement Displacement -0.5 -0.5 0 100 200 0 100 200 0 0.2 0.4 0 0.2 0.4 * * Time t* [1] Time t* [1] Time t [1] Time t [1] * * (d) f u = 0.02 (h) f u = 10
Fig. 4. Nondimensional time response (Nominal model) JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 9
0.01 0.01 a * a c p* p 0.06 + 0.06 + c * [1] 0.005 * 0.005 [1] p d p * ± * ± - [1] - [1] * * b p 0.04 b d p 0.04 b 0 0 c a a d b Input u Input 0.02 u Input 0.02 -0.005 c -0.005 Pressure d Pressure -0.01 0 -0.01 0 0 2000 4000 0 2000 4000 02040 20 40 Time t* [1] Time t* [1] Time t* [1] Time t* [1] 0.5 0.5 c c d [1] 0.01 0.01 [1] * a * s d s b [1] 0.005 [1] 0.005 * c * a a c a d 0 b 0 0 b 0 b d -0.005 -0.005 Force f Force Force f Force -0.01 -0.01 Displacement Displacement -0.5 -0.5 0 2000 4000 0 2000 4000 02040 20 40 * * * * Time t [1] * Time t [1] Time t [1] * Time t [1] (a) f u = 0.001 (e) f u = 0.1
0.01 a a 0.01 * c p c * 0.06 + 0.06 p b + 0.005 [1] * b p 0.005 [1] *
* ± p [1] - * ±
d [1] - * d * p 0.04 p 0.04 0 0 c d b
Input u Input 0.02 -0.005 d u Input -0.005 0.02 a Pressure
a b Pressure c -0.01 0 -0.01 0 0 1000 2000 0 1000 2000 051015 0 5 10 15 * * Time t [1] Time t [1] Time t* [1] Time t* [1]
c 0.5 0.5 d
0.01 [1] c
0.01 [1] * b d a * s s
[1] 0.005
[1] 0.005 * * a c 0 a b 0 0 0 c b d -0.005 d
Force f Force -0.005 Force f Force a -0.01
Displacement -0.01 b Displacement -0.5 -0.5 0 1000 2000 0 1000 2000 051015 0 5 10 15 * * Time t [1] Time t [1] Time t* [1] Time t* [1] * * (b) f u = 0.002 (f) f u = 0.3
0.01 0.01 a a c * p* 0.06 p 0.06 c + +
0.005 [1] * 0.005 [1] * p
p * ± b [1]
* ± -
[1] - * b * p p 0.04 0.04 d d c 0 b 0 d a
Input u Input 0.02 Input u Input -0.005 0.02 b -0.005 Pressure Pressure d a -0.01 0 c -0.01 0 0 500 1000 1500 0 500 1000 1500 05100510 * * Time t* [1] Time t* [1] Time t [1] Time t [1] 0.5 0.5 c
0.01 [1]
0.01 [1] a b * d * c s s
[1] 0.005
[1] 0.005 * * b c a c 0 0 d 0 d 0 a b b d -0.005 -0.005 f Force a Force f Force -0.01
-0.01 Displacement Displacement -0.5 -0.5 0 500 1000 1500 0 500 1000 1500 05100510 * * Time t* [1] Time t* [1] Time t [1] * Time t [1] * (c) f u = 0.003 (g) f u = 0.5
0.01 0.01 a c * a c p* p 0.06 + 0.06 +
[1] *
[1] * 0.005 0.005 p p b * ± * ± [1] - [1] -
* b d * p p 0.04 b 0.04 d d 0 0 a b c d a c
Input u Input 0.02 Input u Input -0.005 0.02 -0.005 Pressure Pressure
-0.01 0 -0.01 0 0 100 200 0 100 200 0 0.2 0.4 0 0.2 0.4 * * Time t* [1] Time t* [1] Time t [1] Time t [1]
c 0.5 0.5
0.01 [1]
0.01 [1] *
d * s s
[1] 0.005 [1] 0.005 c *
* a b a c 0 0 0 0 a b c d a b d b d -0.005 -0.005 Force f Force Force f Force -0.01 -0.01 Displacement Displacement -0.5 -0.5 0 100 200 0 100 200 0 0.2 0.4 0 0.2 0.4 * * Time t* [1] Time t* [1] Time t [1] Time t [1] * * (d) f u = 0.02 (h) f u = 10
Fig. 5. Nondimensional time response (Linearized model) JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 10 took n =10to make Figure 3, the number of updates of the 0.08 0.08 Nominal Nominal physical parameters was remarkably reduced. For the design 0.07 Linearized 0.07 Linearized Linearized (undershoot) optimization [8] which is not our main objective in this paper, 0.06 0.06 since we may need the 5-dimensional search after the 3- 0.05 0.05 dimensional search, the proposed nondimensional representa- [m] [m] tion has O(n3)+O(n5) time complexity at maximum which 0.04 0.04 8 is still better than O(n ) time complexity even at n =2. The 0.03 0.03 Displacement Displacement parameter space reduction was also evaluated. 0.02 0.02
Table 1 evaluates the nonlinear dynamics computation 0.01 0.01 time which is the sum of all computation times within 0 0 ∗ ∗ ∗ D A P space needed to make Figure 3. At every fre- 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ∗ Time t [s] Time t [s] quency fu, as expected, the computation time for x(t)= ∗ ∗ ∗ Xsφ[θspecial(θ),x (0),u (t/Ts)] using the proposed nondi- Fig. 6. Disturbance response (Left: large scale, Right: small scale). mensional representation (3) (4) was better than that for x(t)=φ[θ, x(0),u(t)] using the original representation (1) linear analysis, if F ∗ < {−(2(D∗ − 6(1 + A∗))3/2)/27 + (2). In total, the computation time of the proposed nondimen- √ 2(D∗)3/27 − 2D∗(1 + A∗)/3}/ 8P ∗ ≈ 1.82, the zero- sional representation was reduced to 15710 s (4.2 h) which undershoot is guaranteed for the linearized model (12), but is around the half of that of the original representation 31455 not guaranteed for the nominal model due to the nonlinearity. s (8.7 h). This is because the parameters (M,A+,L,b,C)= Figure 6 shows the disturbance responses (the solid black (1, 1, 1, 1, 1) reduce the number of multiplication and division and red curves) with F =0.0128 (F ∗ =0.908 < 1.82) operations preserving the parametric structure in the original of the nominal model and that of the linearized model (12) representation (1) (2). The computation time can be improved against the step disturbance 20000 N. Fortunately, without more since all existing methods [8] [33] developed for the adjusting F , the zero-undershoot for the nominal model is original representation (1) (2) can be applied. While the ∗ confirmed numerically (not analytically) whereas the response computations were made for only eight frequencies f in Table u (the dashed red curve) with F =0.0345 (F ∗ =2.45 > 1.82) 1, the fast computation was well evaluated. has the nonzero (dangerous) undershoot for the linearized Finally, let us remark that Proposition 1 and Theorem 1 model (12). The response difference between the models with provide the links to the closed-loop discussion as well as the the same gain corresponds to the nonlinearity (the FIT ratio open-loop discussion presented in the above of this paper. ≈ ∗ 0 in the band fu < 0.02) in Figure 3 which justifies to use Remark 3 (Relating to design and control via scaling) Let the nominal model instead of the linearized model (12). us put a simple example of the links based on our experimental [Step 3] In a certain small scale, we design and control 1-DOF arm, and consider a scaling design and control problem the small scaled hydraulic cylinder since the similarity [34] of a hydraulic cylinder whose piston undershoot should be is sometimes required to reduce the experimental cost in zero in the presence of force disturbance. Assume that only the large scale. Here, by replacing M = 100 → 25 and 2 8 ∗ ∗ ∗ ∗ (D, A+,b) = (11000 Ns/m, 0.0021 m , 5.3 × 10 Pa) are keeping (D ,A ,P ) and F ,wehave(M,L,A−,C,P)= 5 −4 6 given and the others (M,L,A−,C,P) ∈ R+ and a gain F>0 (25, 3.6, 0.0016, 1.8 × 10 , 14 × 10 ) and F =0.0512. of the simple control u(t)=−Fs(t) are unknown under a Against the nonlinearity, based on Proposition 1 and Theorem certain working constraints L/2 ≥ 2.5m(with pipeline length 1, the zero-undershoot is guaranteed even for the nominal 2 6 effect), A− ≤ 0.0016 m , P ≤ 21×10 Pa in the large scale. model in the small scale. Indeed, Figure 6 shows no under- [Step 1] In the large scale, we search the parameters shoots in the disturbance responses with F =0.0512 of both (M,L,A−,C,P) by the advanced direct search approach. models in the small scale. Now we can start to construct the Since the transfer function in the following linearization small scaled hydraulic cylinder for the experimental validation. based control (the classical control) does not treat any ini- tial response x¯(t):=φ[θ, x(0),u(t) ≡ 0], the objective V. C ONCLUSION function is the norm overshoot max |x¯(t)TQx¯(t)| by the 0≤t<∞ This paper reveals that a nominal model of hydraulic cylin- random initial state x(0) ∈ ΩLP . Here, we will suffer from ders has a new simplicity on the parametric structure that more O(n5) time complexity without Proposition 1 and Theorem accurate or complex, and merely simple existing models do not 1, but now only O(n3) time complexity is needed. when have. Without loss of generality, only by changing the damping n =10and Q = diag(1, 1, 0.1, 0.1), the searched pa- constant D∗, the rod area A∗, and the source pressure P ∗ rameters are (D∗,A∗,P∗)=(4.1, 0.75, 0.028) which imply and assuming that all the other physical parameters are unity, 2 6 (M,L,A−,C,P)=(M,0.144M,0.0016 m ,C,14×10 Pa) any index, such as the numerical existence and nonlinearity, ∗ ∗ ∗ with the free parameters M and C. Under the constraints, our is visualized efficiently. Three parameters D , A , and P choice is M = 100 kg and C =1.8 × 10−4 m5/kg. The correspond to the damping parameter d ∗ of the mass-damper- physical units are unique and dropped in the following. spring, or to the Froude number Fr and the Reynolds number [Step 2] In the large scale, we prepare the linearization based Re of the fluid system in Section I. This is an inevitable, control u(t)=−Fs(t) whose nondimensional version is unexpected, and economical result. Roughly speaking, Figure ∗ ∗ ∗ ∗ ∗ ∗ ∗ u (t )=−(1/Us)F (Ls (t )) =: F s (t ). By the standard 4 is the best possible result corresponding to the analytical JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 11
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Detection and isolation of leakage and valve faults in hydraulic systems in varying loading condition, part 1: Global Satoru Sakai Satoru Sakai received the B.E. degree in Engineering and the M.E. and Ph.D. degrees sensitivity analysis. International Journal of Fluid Power, 12:41-51, 2011. in Agricultural Engineering from Kyoto University, [11] M. Vilenius. Application of sensitivity analysis to electrohydraulic Japan, in 1998, 2000 and 2003, respectively. In PLACE position servos. ASME Journal of Dynamic Systems, Measurements and 2005, he joined Chiba University. In 2010, he joined PHOTO Shinshu University, Japan, where he is currently an Control, 106:77-82, 1984. HERE associate professor with the Department of Mechan- [12] G. Grabmair and K. Schlacher. Energy-based nonlinear control of ical Engineering. From 2003 to 2005, he was a JSPS hydraulically actuated mechanical systems. In Proc. of IEEE CDC, pages postdoctoral research fellow with the Department of Informatics, Kyoto University, and from 2004 to 7520-7525, 2005. 2005 he was a visiting researcher with the Faculty [13] A. Kugi and W. Kemmetmuller. New energy-based nonlinear controller of Electrical Engineering, University of Twente, Netherlands. His research & for hydraulic piston actuators. European Journal of Control, 10:163-173, 2004. interests include robotics, systems control. He was a recipient of the Young Author Prize at IFAC 2005. [14] P. Y. Li and M. R. Wang. Natural storage function for passivity- based trajectory control of hydraulic actuators. IEEE/ASME Transactions on Mechatoronics, 19(3):1057-1068, 2014. [15] S. Sakai and Y. Maeshima. A new method for parameter identification Stefano Stramigioli Stefano Stramigioli received for N-dof hydraulic robots. In Proc. of IEEE ICRA, pages 5983-5989, 2014. the M.Sc. (Hons.) degree (cum laude) from the [16] K. Zhou, J. C. Doyle, and K. Glover. Robust and Optimal Control. University of Bologna, Bologna, Italy, in 1992 and Prentice-Hall, 1996. the Ph.D. (Hons.) degree (cum laude) from the Delft PLACE University of Technology, Delft, The Netherlands, in [17] A. J. van der Schaft. L2-Gain and Passivity Techniques in Nonlinear PHOTO 1998. He is currently a Full Professor of advanced Control. Springer-Verlag, 2000. HERE robotics and the chair holder of the Robotics and [18] E. Buckingham. On physically similar systems. Physics review, (4):354- Mechatronics Group, University of Twente. He has more than 200 publications including four books. 376, 1931. He is currently the Vice-President for Research of [19] W. D. Curtis, J. David Logan, and W. A. Parker. Dimensional analysis euRobotics, the private part of the PPP with the and the pi theorem. Linear Algebra and Its Applications, (47):117-126, 1982. EU known as SPARC. He has been an Officer and AdCom Member for IEEE/RAS. [20] K. Fujimoto and T. Sugie. Canonical transformation and stabilization