JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 1, pp. 163-184, Warsaw 2006 PIECEWISE LINEAR luz(. . .) AND tar(. . .) PROJECTIONS. PART 1 – THEORETICAL BACKGROUND Dariusz Żardecki Automotive Industry Institute (PIMOT), Warsaw e-mail: zardecki@pimot.org.pl The paper presents definitions and theorems for luz(. . .) and tar(. . .) piecewise linear projections. These projections and their originalmathe- matical apparatus are very useful formodelling of nonlinear systems, eg systems with freeplay or friction. Key words: non-linearties, piecewise linear systems, algebraic and diffe- rential equations 1. Introduction Nonlinear systemswhich can bemodelled using piecewise linear equations are called ”piecewise linear systems”. Oftentimes, the piecewise linearity is a result of non-linear approximation of a function. But piecewise linear charac- teristics with well-marked points of ”fractures” can be consequences of varia- tional principles referring to physical processes with constrains – see Grzesi- kiewicz (1990) for examples. So, from the mathematical point of view, such a piecewise linearity might be a result of some optimization task with limits (see example below). Theorem 1.1. Theoptimisation task yopt(x,a) : minQy(x−y)∧y∈ [−a,a], where Q(x−y) is a convex function has the solution yopt(x,a) =      a if x­ a x if −a 0 166 D.Żardecki Fig. 2. Geometric interpretation of luz(. . .) and tar(. . .) projections Thecross-invertibility of the luz(. . .) and tar(. . .) projection is theirmain attribute. Theorem 2.1. (On invertibility, formal proof by Żardecki (2001)) luz(x,a)= tar−1(x,a) tar(x,a)= luz−1(x,a) For all x, the luz(x,a) and tar(x,a) are like anti-functions, when a has a non-negative value. Note that such attribute is not true (see Fig.3) for projections luz(x,−a) and tar(x,−a) defined according to the presented formulas (details by Żardecki (2001)). Fig. 3. Geometric interpretation of projections with a negative parameter The luz(. . .) and tar(. . .) can be treated as cases of a more general talu(x,a1,a2) projection. Definition 2.2. For −∞a x if b= a tar(x,a− b) if ba1 (k1±k2) luz(x,a1) if a2 = a1 ±k2[ luz(x,a2)− luz(x,a1)]+(k1±k2) luz(x,a1) if a2 0 (k1±k2)x if k1a1±k2a2 k1±k2 =0 (k1±k2) [ 2x− tar ( x, ∣ ∣ ∣ k1a1±k2a2 k1±k2 ∣ ∣ ∣ )] if k1a1±k2a2 k1±k2 < 0 Note: In the case of summation, this formula simplifies to a compact formula k1 tar(x,a1)+k2 tar(x,a2)= (k1+k2)tar ( x, k1a1+k2a2 k1+k2 ) Theorem 3.6. (On disentanglement of the feedback system with luz(. . .)) If luz(y,b) = k luz(x−y,a) then luz(y,b)= k k+1 luz(x,a+ b) luz(x−y,a)= luz(x,a+ b)− luz(y,b) y= k k+1 talu ( x,a+ b, k+1 k b ) x= k+1 k talu ( y,b, k k+1 (a+ b) ) luz(y,b) k→∞ −→ luz(x,a+ b) Note: For linear system (when a= b=0) it means self-evident dependence: If y= k(x−y) then y= k k+1 x From Theorem 3.6 we can create another formulas, for example: If y= k luz(x−y,a)+ c then y= k k+1 luz(x− c,a)+ c and so on. 170 D.Żardecki Theorem 3.7. (On disentanglement of the feedback system with tar(. . .)) If tar(y,b)= k tar(x−y,a) then y=                x− 1 k+1 luz(x,ka− b) if ka> b k k+1 x if ka= b k k+1 luz ( x, b−ka k ) if ka< b x=              y+ 1 k luz(y,ka− b) if ka> b ( 1+ 1 k ) y if ka= b ( 1+ 1 k ) tar ( y, b−ka k+1 ) if ka< b y k→∞ −→ x Theorem 3.8. (On disentanglement of the feedback system with luz(. . .) and tar(. . .)) If luz(y,b) = k tar(x−y,a) then y=x− 1 k+1 luz(x,ka+b) x= y+ 1 k luz(y,ka+b) y k→∞ −→ x Theorem 3.9. (On disentanglement of the feedback system with tar(. . .) and luz(. . .)) If tar(y,b)= k luz(x−y,a) then y= k k+1 luz ( x, ka+ b k+1 ) x= k+1 k tar ( y, ka+ b k+1 ) y k→∞ −→ luz(x,a) The main advantage of the elaborated mathematical apparatus for piece- wise linear systems is the possibility of finding rather simple mathematical dependences. Formulas concerning algebraic operations are analogous to well known formulas of standard linear systems. The mathematical apparatus of luz(. . .) and tar(. . .) coheres also with topological procedures basing on gra- phs or block diagrams. For example, Theorem 3.6 expressed by block-diagram symbols is illustrated in Fig.5. Piecewise linear luz(. . .) and tar(. . .) projections. Part 1... 171 Fig. 5. Block-diagram interpretation of Theorem 3.6 The analytical formulas enable reduction of cascade piecewise systems. For example tar(luz(luz(x,a),b),c) = tar(luz(x,a+ b),c)= talu(x,a+b,c) The formulas concerningdisentanglement of feedback systemwith luz(. . .) or/and tar(. . .))projections enables transformationandsimplificationof com- plex models governed by piecewise linear algebraic equations. This important matter will be discussed in the next section. 4. Algebraic equations with luz(. . .) and tar(. . .) Oftentimes, in multi-dimensional piecewise linear systems, output varia- bles are not explicity dependent on input variables, and they are liable to constraints given by involved piecewise linear algebraic equations (treated as static subsystems). The problem of their clearing turns out to be very impor- tant for effective numerical simulation. If such piecewise linear constraints are composed of luz(. . .) and tar(. . .) projections, an analytical disentanglement may be unexpectedly easy to carry out. The basic mathematical apparatus can be applied directly (for example Theorem 3.6) to one-dimensional con- straint equations. For two-dimensional equations, theorems presented below are a new chance. Attention: constants a,b,c,k1,k2 in the following theorems are non-negative. 172 D.Żardecki Theorem 4.1. If y+k1 luz(y−x,a)= f x−k2 luz(y−x,a)= g then y= f− k1 k1+k2+1 luz(f−g,a) x= g+ k2 k1+k2+1 luz(f−g,a) Proof From the first equation 1 k1 (f−y)= luz(y−x,a) hence x= y− tar (f−y k1 ,a ) From the first and second equations f−y k1 = x−g k2 or x= k2 k1 (f−y)+g hence y− tar (f−y k1 ,a ) = k2 k1 (f−y)+g y−g= 1 k1 tar(f−y,k1a)+ k2 k1 (f−y) On the basis of Theorem 3.5 y−g= ( 1 k1 + k2 k1 ) tar ( f−y, k1a 1 k1 + k2 k1 ) hence luz(y−g,k1a)= 1+k2 k1 (f−y) f−y= k1 k2+1 luz(f−g− (f−y),k1a) FromTheorem 3.6 f−y= k1 k2+1 k1 k2+1 +1 luz(f−g,k1a) hence y= f− k1 k2+k1+1 luz(f−g,a) x= k2 k1 ( f−f+ k1 k2+k1+1 luz(f−g,a) ) +g Piecewise linear luz(. . .) and tar(. . .) projections. Part 1... 173 and finally x= g+ k2 k2+k1+1 luz(f−g,a) � Note: If k1 = k, k2 = pk (linear dependence k1 and k2), then y k→∞ −→ f− 1 p+1 luz(f−g,a) x k→∞ −→ g+ p p+1 luz(f−g,a) Theorem 4.2. If y+k1 tar(y−x,a)= f x−k2 tar(y−x,a)= g then y= k2f+k1g k2+k1 + k1 k2+k1+1 luz ( f−g k2+k1 ,a ) x= k2f+k1g k2+k1 − k2 k2+k1+1 luz ( f−g k2+k1 ,a ) Proof From the first equation f−y k1 = tar(y−x,a) hence x= y− luz (f−y k1 ,a ) From the first and second equation f−y k1 = x−g k2 or x= k2 k1 (f−y)+g hence y− luz (f−y k1 ,a ) = k2 k1 (f−y)+g k1y−k2(f−y)−k1g= luz(f−y,k1a) (k1+k2)y− (k2f+k1g)= luz(f−y,k1a) y− k2f+k1g k1+k2 = 1 k1+k2 luz(f−y,k1a) also y− k2f+k1g k1+k2 = 1 k1+k2 luz ( f− k2f+k1g k1+k2 − ( y− k2f+k1g k1+k2 ) ,k1a ) 174 D.Żardecki On the basis of Theorem 3.6 y− k2f+k1g k1+k2 = 1 k1+k2 1 k1+k2 +1 luz ( f− k2f+k1g k1+k2 ,k1a ) y− k2f+k1g k1+k2 = 1 k1+k2+1 luz (k1(f−g) k1+k2 ,k1a ) hence y= k2f+k1g k2+k1 + k1 k2+k1+1 luz ( f−g k2+k1 ,a ) So x= k2 k1 ( f− k2f+k1g k2+k1 − k1 k2+k1+1 luz ( f−g k1+k2 ,a )) +g hence x= k2f+k1g k2+k1 − k2 k2+k1+1 luz ( f−g k2+k1 ,a ) � Note: If k1 = k, k2 = pk (linear dependence k1 and k2), then y k→∞ −→ pf+g p+1 x k→∞ −→ pf+g p+1 The variables f and g can be treated as input variables and y and x – as output ones for two-dimensional static systems that were initialy described by equations with entangled outputs. Those entanglements disappear thanks to the presented theorems. Sometimes, as a result of mathematical modelling, one obtains some mul- tidimensional model with redundant variables. The theorem presented below might be very useful for analytical reduction of the model. Theorem 4.3. If luz(y−w,a) = k1 luz(w−u,c) luz(u−x,b)= k2 luz(w−u,c) then luz(y−w,a) = k1 k2+k1+1 luz(y−x,a+b+ c) luz(u−x,b)= k2 k2+k1+1 luz(y−x,a+ b+ c) Piecewise linear luz(. . .) and tar(. . .) projections. Part 1... 175 Proof From the first equation, we have u=w− tar ( 1 k1 luz(y−w,a),c ) On the basis of both equations luz(u−x,b)= k2 k1 luz(w−u,c) hence luz ( w−x− tar ( 1 k1 luz(y−w,a),c ) ,b ) = k2 k1 luz(y−w,a) After inversion w−x− tar ( 1 k1 luz(y−w,a),c ) = tar (k2 k1 luz(y−w,a),b ) hence w−x= 1 k1 tar(luz(y−w,a),k1c)+ k2 k1 tar ( luz(y−w,a), k1 k2 b ) On the basis of Theorem 3.5 w−x= ( 1 k1 + k2 k1 ) tar ( luz(y−w,a), c+ b 1 k1 + k2 k1 ) hence w−x= tar (1+k2 k1 luz(y−w,a),b+ c ) luz(w−x,b+ c)= 1+k2 k1 luz(y−w,a) luz(y−w,a)= k1 k2+1 luz(w−x,b+ c) or luz(y−w,a)= k1 k2+1 luz(y−x− (y−w),b+ c) On the basis of Theorem 3.7 luz(y−w,a)= k1 k2+1 k1 k2+1 +1 luz(y−x,a+ b+ c) hence luz(y−w,a)= k1 k2+k1+1 luz(y−x,a+ b+ c) 176 D.Żardecki The proof of the first part of theorem is ended. The proof of the second part runs similarly. From the second equation w=u+ tar ( 1 k2 luz(u−x,b),c ) Taking into account that luz(y−w,a)= k1 k2 luz(u−x,b) luz ( y−u− tar ( 1 k2 luz(u−x,b),c ) ,a ) = k1 k2 luz(u−x,b) y−u− tar ( 1 k2 luz(u−x,b),c ) = tar (k1 k2 luz(u−x,b),a ) hence y−u= tar (k1+1 k2 luz(u−x,b),a+ c ) luz(u−x,b)= k2 k1+k2 luz(y−u,a+ c) luz(u−x,b)= k2 k1+k2 luz(y−x− (u−x),a+ c) On the basis of Theorem 3.7 also luz(u−x,b)= k2 k1+k2+1 luz(y−x,a+ b+ c) � Note: If k1 = k, k2 = pk (linear dependence), then luz(y−w,a) k→∞ −→ 1 p+1 luz(y−x,a+b+ c) luz(u−x,b) k→∞ −→ p p+1 luz(y−x,a+ b+ c) There is possible a formulation of analogous theorems for systems of equ- ations with constraints containing the tar(. . .) or mixed pair luz(. . .) and tar(. . .) projections. Such analytical formulas have rather complicated forms. In comments to Theorems 4.1-4.3 we have considered also peculiar cases when the coefficients k1,k2 were extremely large, but linearly dependent. Such anoutwardly impossible situation takes place in the casewhenamathematical model of so the called stiff dynamic system is set. For stiff systems, degene- ration of equations of motion can be done by parametric operations. Such a simplification of the model is easy to execute using the proved formulas for disentanglement. Piecewise linear luz(. . .) and tar(. . .) projections. Part 1... 177 5. Basic properties of ordinary differential equations and inclusions with luz(. . .) and tar(. . .) projections In this Section, we investigate basicmathematical properties of dynamical systems described by equations and inclusionswith the luz(. . .) and tar(. . .) projections. The necessity of taking into account not only equations but also inclusions result from the indetermination (even though at the beginning of our study) of tar(0,a). So, since tar(0,a) = as∗ ∈ [−a,a]): • instead of formula ẋ(t)= f(. . . , tar(x(t),a), . . .) (differential state equ- ation) we have ẋ(t) ∈ f(. . . , tar(x(t),a), . . .) (differential state inclu- sion), • instead of formula 0 = f(. . . , tar(x(t),a), . . .) (function equation) we have 0∈ f(. . . , tar(x(t),a), . . .) (function inclusion). Transformation of inclusion description requires an individual approach. In cases when the tar(. . .) projections are elements of a single inclusion, the theorems presented below can be very useful. Attention: parameters a,b, . . . appearing in the following theorems are non- negative. Theorem 5.1. Inclusion ẋ(t)∈ y(t)− btar(x(t),a) for which tar(0,a) per- forms the optimization task tar(0,a)opt : min tar(0,a) Q(ẋ) ∧ tar(0,a)∈ [−a,a] where Q(. . .) is a convex function is equivalent to: – differential equation with singularity s∗ ∈ [−1,1] ẋ(t)= y(t)− btar(x(t),a) where bas∗ = b(tar(0,a))opt = y(t)− luz(y(t),ba) – differential variable-structure equation ẋ(t)= { y(t)− btar(x(t),a) if x(t) 6=0 luz(y(t),ba) if x(t)= 0 178 D.Żardecki Proof The differential inclusion ẋ(t)∈ y(t)− tar(x(t),a) is equivalent to the differential equation ẋ(t)= { y(t)− btar(x(t),a) if x(t) 6=0 y(t)− btar(0,a) if x(t)= 0 Because of tar(0,a)= as∗, where s∗ ∈ [−1,1], we have btar(0,a)∈ [−ba,ba]. On the basis of Theorem 1.1, applying luz(. . .) notation, the task b(tar(0,a))opt : min btar(0,a) Q(ẋ(t))= min btar(0,a) Q(y(t)− btar(0,a)) ∧ btar(0,a)∈ [−ba,ba] has the solution b(tar(0,a))opt = y(t)− luz(y(t),ba). Therefore, also ẋ(t)= { y(t)− btar(x(t),a) if x(t) 6=0 y(t)− b(tar(0,a))opt = luz(y(t),ba) if x(t)= 0 � Both forms of the description are equivalent. Determination of tar(0,a) in the optimization task b(tar(0,a))opt = y(t)− luz(y(t),ba) caused a new situation in which for x(t)= 0 themacro-projection btar(x,a) is replaced by a new piecewise linear macro-projection based on the variable y(t) and the luz(. . .) projection. Fig. 6. Determination of tar(x,a) projection for x=0 Finally, for x(t) = 0, the state equation has been described as ẋ(t) = = luz(y(t),ba). Analysing this form, we ascertain that it express a practical rule: ”for x(t)= 0 if y(t)∈ [−ba,ba] the blocked state (ẋ(t)= 0) is held as far as y(t) /∈ [−ba,ba]”. The calculation of b(tar(0,a))opt = y(t)− luz(y(t),ba) on the basis of formal minimization Q(ẋ(t)) is equivalent with application of the heuristic rule describing ”motion blockade”. Such replacement of the for- mal approach (optimization) by the well known heuristic rule is an important practical method for resolving inclusion problems. Piecewise linear luz(. . .) and tar(. . .) projections. Part 1... 179 Theorem 5.2. The inclusion 0 ∈ y(t)− btar(x(t),a) is equivalent to the equation x(t)= luz (1 b y(t),a ) Proof The inclusion 0∈ y(t)− btar(x(t),a) is equivalent to 0= { y(t)− btar(x(t),a) if x(t) 6=0 y(t)− btar(0,a) if x(t)=0 For x(t) 6=0 from y(t)− btar(x(t),a)= 0, we obtain luz (y(t) b ,a ) =x(t) For x(t)= 0 from y(t)− btar(0,a)=0, we obtain y(t) b =0=x(t) or luz (y(t) b ,a ) =0=x(t) So, in fact x(t)= luz (y(t) b ,a ) for all x(t). � Theorem 5.3. Degeneration of the inclusion εẋ(t)∈ y(t)− btar(x(t),a) by ε→ 0 gives the equation x(t)= luz (y(t) b ,a ) Proof εẋ(t)∈ y(t)− btar(x(t),a) ε→0 −→ 0∈ y(t)− btar(x(t),a) On the basis of Theorem 6.2, we obtain the final result. � On the basis of Theorem5.3, we conclude that parametric reduction of the inclusive model is deprived of its ambiguousness. When the argument of the tar(. . .) projection is given by a linear combi- nation of variables, Theorems 5.1-5.3 can be used directly, but for a modified formof the inclusionmodel. For a typical two-variablemodel, this is presented in the proof of Theorem 5.4. 180 D.Żardecki Theorem 5.4. The inclusion [ ẋ1(t) ẋ2(t) ] ∈ [ y1(t)− b1 tar(x1(t)−x2(t),a) y2(t)+ b2 tar(x1(t)−x2(t),a) ] for which tar(0,a) performs the optimization task tar(0,a)opt : min tar(0,a) Q(ẋ1(t)− ẋ2(t)) ∧ tar(0,a)∈ [−a,a] Q(. . .) is a convex function is equivalent to: – differential equation with singularity s∗12 ∈ [−1,1] [ ẋ1(t) ẋ2(t) ] = [ y1(t)− b1 tar(x1(t)−x2(t),a) y2(t)+ b2 tar(x1(t)−x2(t),a) ] where as∗12 = tar(0,a)opt = = 1 b1− b2 [y1(t)−y2(t)− luz(y1(t)−y2(t),(b1− b2)a)] – differential variable-structure equation [ ẋ1(t) ẋ2(t) ] = [ y1(t)− b1 tar(x1(t)−x2(t),a) y2(t)+ b2 tar(x1(t)−x2(t),a) ] if x1(t) 6=x2(t) [ ẋ1(t) ẋ2(t) ] =     y1(t)− b1 b1− b2 [y1(t)−y2(t)− luz(y1(t)−y2(t),(b1− b2)a)] y2(t)+ b2 b1− b2 [y1(t)−y2(t)− luz(y1(t)−y2(t),(b1− b2)a)]     if x1(t)=x2(t) Proof We create a new equation for the variable x12(t)=x1(t)−x2(t). Subtracting the state equations ẋ12(t)= y1(t)−y2(t)− (b1− b2)tar(x(t),a) For x12(t)=x1(t)−x2(t)= 0 using Theorem 6.1, we obtain (b1− b2)tar(0,a)opt =(b1− b2)as ∗ 12 = y1(t)−y2(t)− luz(y1(t)−y2(t),(b1− b2)a) hence tar(0,a)opt = 1 b1− b2 [y1(t)−y2(t)− luz(y1(t)−y2(t),a)] After substitution we obtain final results. � Piecewise linear luz(. . .) and tar(. . .) projections. Part 1... 181 In more complicated cases, when model equations contain multiple com- ponents tar(xi,ai) and tar(xi −xj,aij), creation of new inclusions for the variables xij = xi − xj leads to a unified multidimensional form of the model [ẋ1(t), ẋ2(t), . . . , ẋn(t)] > ∈ ∈        y1(t)− b11 tar(x1(t),a1)− b12 tar(x2(t),a2)+ . . .−b1n tar(xn(t),an) y2(t)− b21 tar(x1(t),a1)− b22 tar(x2(t),a2)+ . . .−b2n tar(xn(t),an) ... yn(t)− bn1 tar(x1(t),a1)− bn2 tar(x2(t),a2)+ . . .− bnn tar(xn(t),an)        Such amodel should be completed by a rule of calculation of the unknown tar(0,ai), eg. theoptimization taskbasedongeneral physical principles.But in many cases, calculation of tar(0,ai) canbe resolvedpractically usingheuristic procedures.Suchaprocedure (here theS-Sprocedure) fordescriptionof the so- called ”stick-slip” process in themultidimensionalmodel is presentedbelow. It makes use of the fact that for xi(t)= 0amotion is blocked (ẋi(t)= 0) only for s∗i(t)∈ [−1,1]. The saturation formula s ∗ i(t) = s ∗∗ i (t)− luz(s ∗∗ i (t),1) expres- sing the limitations on s∗i(t), where s ∗∗ i (t) refers to blocked state, enables description of such ”stick-slip” conditions in the state x1(t)= 0, x2(t)= 0,..., xn(t)= 0. Definition 5.1. The S-S procedure for disentanglement of the inclusion sys- tem and calculation of the ”stick-slip” process: 1. Determination of the ”stick-slip” variables xi for which xi(tk)= 0 at t= tk 2. Determination of the ”stick-slip” subsystem of the equations ẋi(tk)= . . . 3. Setting tar(xi(t),ai)|xi(t)=0 = ais ∗ i = a(s ∗∗ i − luz(s ∗∗ i ,1)) in the ”stick-slip” subsystem 4. Calculation of s∗∗i from the ”stick-slip” subsystem for ẋi(tk)= 0 5. Calculation of tar(xi(t),ai)|xi(t)=0 = ais ∗ i = a(s ∗∗ i − luz(s ∗∗ i ,1)) 6. Calculation of ẋi(tk) from the system equations. Note that applying the S-Sprocedure, for the state x1(t)= 0,x2(t)= 0,..., xn(t)= 0, we obtain 182 D.Żardecki       0 0 ... 0       =       y1(t)− b11a1s∗∗1 − b12a2s ∗∗ 2 + . . .−b1nans ∗∗ n y2(t)− b21a1s∗∗1 − b22a2s ∗∗ 2 + . . .−b2nans ∗∗ n ... yn(t)−bn1a1s∗∗1 − bn2a2s ∗∗ 2 + . . .− bnnans ∗∗ n       [ẋ1(t), ẋ2(t), . . . , ẋn(t)] > = =       y1(t)− b11a1[s∗∗1 (t)− luz(s ∗∗ 1 (t),1)]− b12a2[s ∗∗ 2 (t)− luz(s ∗∗ 2 (t),1)]+ . . . y1(t)− b21a1[s∗∗s (t)− luz(s ∗∗ 1 (t),1)]− b22a2[s ∗∗ 2 (t)− luz(s ∗∗ 2 (t),1)]+ . . . ... yn(t)− bn1a1[s∗∗1 (t)− luz(s ∗∗ 1 (t),1)]− bn2a2[s ∗∗ 2 (t)− luz(s ∗∗ 2 (t),1)]+ . . .       and finally [ẋ1(t), ẋ2(t), . . . , ẋn] > = =       −b11a1 luz(s∗∗1 (t),1)− b12a2 luz(s ∗∗ 2 (t),1)+ . . .− b1nan luz(s ∗∗ n (t),1) −b21a1 luz(s∗∗1 (t),1)− b22a2 luz(s ∗∗ 2 (t),1)+ . . .− b2nan luz(s ∗∗ n (t),1) ... −bn1a1 luz(s∗∗1 (t),1)−bn2a2 luz(s ∗∗ 2 (t),1)+ . . .− bnnan luz(s ∗∗ n (t),1)       If for all i = 1,2, . . . ,n the calculated s∗∗i ∈ [−1,1], the all luz(s ∗∗ i ,1) = 0 and all ẋi(t) = 0. It means total blockade of the system. If the calculated s∗∗i /∈ [−1,1], then luz(s ∗∗ i ,1) 6= 0 as well. Thus the right sides of the equ- ations are non-zero wherethrough the blockade state of that variables can be terminated. Obviously, this way one can concern only some variables. As a re- sult of the standardODE (OrdinaryDifferential Equation) solver procedure, a newdynamic state is calculated. This new statemay contain new singularities caused by some xi(t)= 0. The S-S procedure is used once again. For full analysis, the S-S procedure should be applied alternately and in- dependently for every combination of the singularity states, i.e.: – for x1(t)= 0, x2(t) 6=0, x3(t) 6=0,..., xn(t) 6=0 – for x1(t) 6=0, x2(t)= 0, x3(t) 6=0,..., xn(t) 6=0 ... – for x1(t)= 0, x2(t)= 0, x3(t) 6=0,..., xn(t) 6=0 ... – for x1(t)= 0, x2(t)= 0, x3(t)= 0,..., xn(t) 6=0 ... – for x1(t)= 0, x2(t)= 0, x3(t)= 0,..., xn(t)= 0 Piecewise linear luz(. . .) and tar(. . .) projections. Part 1... 183 Finally, we obtainmulti-structural equationswhich take into consideration all singular situations. Such a model can be complicated but ready to use in simulations. The S-S procedure seems to be an attractive proposition for solving the in- clusionproblemswhichappear inmulti-bodymechanical systemswithblocked motion, for example – inmechanismswithmultiple dry friction (stick-slip pro- blems).Obviously, in such cases, the S-Sprocedure leads to the same results as the formal solution to the optimization task based on theGauss ”acceleration energy function”. This is described in the second part of the paper. 6. Final remarks The paper presents the concept, definitions and theorems concerning the luz(. . .) and tar(. . .) projections. These piecewise linear projections have very interesting mathematical properties. Basic formulas, eg. on compounds, line- ar combination, disentanglement of feedback systems constitute surprisingly simple ”algebra” apparatus.The theorems aswell as theS-Sprocedure concer- ning differential inclusions enable efficient analysis of piecewise linear dynamic systems described with the luz(. . .) and tar(. . .). The luz(. . .) and tar(. . .) projections seem to be an interesting idea for investigatorsworkingonpiecewise linearmodels.Applications concerningnon- linearmechanical systemswith freeplay (backlash, clearance) and friction (Co- ulomb’s fiction with stiction) are discussed in the second part of the paper (Żardecki, 2006). Acknowledgments This work has been supported by grants 9T12C07108, 9T12C05819 and 4T07B05928 (a project sponsored by theMinistry of Science and Informatics in 2005- 2007). References 1. Grzesikiewicz W., 1990, Dynamics of mechanical systems with constraints, Prace Naukowe Politechniki Warszawskiej. Mechanika, 117 [in Polish] 2. Kevenaar T.A.M., Leenaerts D.M.W., 1992, A comparison of piecewise- linear model descriptions, IEEE Transactions on Circuit and Systems, 39, 12 184 D.Żardecki 3. Żardecki D., 1992, Structural sensitivity analysis of mathematical models of dynamical systems with application to car-steerability model, Dissertation: WAT,Warszawa [in Polish] 4. Żardecki D., 1995, Themethod of notation and computation of rigidity cha- racteristics of the nonlinear spring systems,Proceedings of Conference AUTO- PROGRES’95, Jachranka 1995, PIMOT, [in Polish] 5. Żardecki D., 2001, The luz(. . .) and tar(. . .) projections – a theoretical background and an idea of application in a modeling of discrete mechanical systems with backlashes or frictions,Biuletyn WAT,L, 5 [in Polish] 6. Żardecki D., 2005, Piecewise-linear modeling of dynamic systems with fre- eplay and friction, Proceedings of 8th DSTA Conference, Łódź, TU of Łódź 7. ŻardeckiD., 2006,Piecewise linear luz(. . .) and tar(. . .) projection.Part2– Application inmodellingof dynamic systemswith freeplayand friction,Journal of Theoretical and Applied Mechanics, 44, 1 Przedziałami liniowe odwzorowania luz(. . .) i tar(. . .). Część 1 – Podstawy teoretyczne Streszczenie Artykuł przedstawia definicje i twierdzenia dotyczące przedziałami liniowych od- wzorowan luz(. . .) i tar(. . .). Odwzorowania i ich oryginalny aparat matematyczny są bardzo użyteczne dla modelowania układów nieliniowych, np. układów z luzem i tarciem. Manuscript received May 13, 2005; accepted for print November 10, 2005