Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j(Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 2: Differential Equations As System Models Ordinary differential equations(ODE)are the most frequently used tool for modeling continuous-time nonlinear dynamical systems. This section presens results on existence of solutions for ODE models, which, in a systems context, translate into ways of proving well-posedness of interconnections 2.1 odE models and their solutions Ordinary differential equations are used to describe responses of a dynamical system to all possible inputs and initial conditions. Equations which do not have a solution for some valid inputs and initial conditions do not define system's behavior completely, and, hence are inappropriate for use in analysis and design. This is the reason a special attention is paid in this lecture to the general question of existence of solution of differential equation 2.1.1 ODE and their solutions An ordinary differential equation on a subset Z R is defined by a function a:ZHR". Let T be a non-empty convex subset of R(i.e. T can be a single point set, or an open, closed, or semi-open interval in R). A function t: TH+R is called a solution of the ODe (t)=a(x(t),t) if(x(t),t)∈ Z for all t∈T,and r(t2)-x(t1)=/a(a(t), t)dt Vti, t2 ET 2 Version of September 10, 2003
� Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 2: Differential Equations As System Models1 Ordinary differential requations (ODE) are the most frequently used tool for modeling continuous-time nonlinear dynamical systems. This section presens results on existence of solutions for ODE models, which, in a systems context, translate into ways of proving well-posedness of interconnections. 2.1 ODE models and their solutions Ordinary differential equations are used to describe responses of a dynamical system to all possible inputs and initial conditions. Equations which do not have a solution for some valid inputs and initial conditions do not define system’s behavior completely, and, hence, are inappropriate for use in analysis and design. This is the reason a special attention is paid in this lecture to the general question of existence of solution of differential equation. 2.1.1 ODE and their solutions An ordinary differential equation on a subset Z � Rn × R is defined by a function a : Z ∈� Rn. Let T be a non-empty convex subset of R (i.e. T can be a single point set, or an open, closed, or semi-open interval in R). A function x : T ∈� Rn is called a solution of the ODE x˙ (t) = a(x(t), t) (2.1) if (x(t), t) ⊂ Z for all t ⊂ T, and t2 x(t2) − x(t1) = a(x(t), t)dt � t1, t2 ⊂ T. (2.2) t1 1Version of September 10, 2003
The variable t is usually referred to as the"time Note the use of an integral form in the formal definition(2.2): it assumes that the function tHa(a(t), t)is integrable on T, but does not require =a(t)to be differentiable at any particular point, which turns out to be convenient for working with discontinuous input signals, such as steps, rectangular impulses, etc Example 2.1 Let sgn denote the "sign "function sgn: R-+0,-1, 1) defined by >0. sgn(y)=0,y=0. 1,y<0. The notation (23) hich can be thought of as representing the action of an on off negative feedback(o describing behavior of velocity subject to dry friction), refers to a differential equation defined as above with n=1,Z=RR(since sgn(a) is defined for all real a, and no restrictions on r or the time variable are explicitly imposed in(2.3), and a(a, t)=sgn(a) It can be verified that all solutions of (2.3) have the form max Ic-t, o or (t)=min(t-c, 01, where c is an arbitrary real constant. These solutions are not differentiable at the critical “ stopping moment”t=c 2.1.2 Standard ODE system models Ordinary differential equations can be used in many ways for modeling of dynamical systems. The notion of a standard OdE system model describes the most straightforward way of doing this Definition A standard ODE model B=OdE(, g) of a system with input u=u(t)E V CR and output w(t)E w CR is defined by a subset X C R", two functions f:X×V×R+→R"andg:X×V×R+→W, and a subset X0CX, so that the behavior set B of the system consists of all pairs(u, a)of signals such that u(t)E for all t, and there exist a solution R+bX of the differential equation i(t)=f(ar(t), v(t), t) such that r(0)∈ Xo and (1)=9(x(t),v(t),t) (25) A special case of this definition, when the input v is not present, defines an autonomous 2Do it as an excercise
� � 2 The variable t is usually referred to as the “time”. Note the use of an integral form in the formal definition (2.2): it assumes that the function t ∈� a(x(t), t) is integrable on T, but does not require x = x(t) to be differentiable at any particular point, which turns out to be convenient for working with discontinuous input signals, such as steps, rectangular impulses, etc. Example 2.1 Let sgn denote the “sign” function sgn : R � {0, −1, 1} defined by � 1, y > 0, sgn(y) = 0, y = 0, −1, y < 0. The notation x˙ = −sgn(x), (2.3) which can be thought of as representing the action of an on/off negative feedback (or describing behavior of velocity subject to dry friction), refers to a differential equation defined as above with n = 1, Z = R × R (since sgn(x) is defined for all real x, and no restrictions on x or the time variable are explicitly imposed in (2.3)), and a(x, t) = sgn(x). It can be verified2 that all solutions of (2.3) have the form x(t) = max{c − t, 0} or x(t) = min{t − c, 0}, where c is an arbitrary real constant. These solutions are not differentiable at the critical “stopping moment” t = c. 2.1.2 Standard ODE system models Ordinary differential equations can be used in many ways for modeling of dynamical systems. The notion of a standard ODE system model describes the most straightforward way of doing this. Definition A standard ODE model B = ODE(f, g) of a system with input v = v(t) ⊂ V � Rm and output w(t) ⊂ W � Rk is defined by a subset X � Rn, two functions f : X × V × R+ ∈� Rn and g : X × V × R+ ∈� W, and a subset X0 � X, so that the behavior set B of the system consists of all pairs (v, w) of signals such that v(t) ⊂ V for all t, and there exist a solution x : R+ ∈� X of the differential equation x˙ (t) = f(x(t), v(t), t) (2.4) such that x(0) ⊂ X0 and w(t) = g(x(t), v(t), t). (2.5) A special case of this definition, when the input v is not present, defines an autonomous system. 2Do it as an excercise!
2.1.3 Well-posedness of standard OdE system models As it was mentioned before, not all ODE models are adequate for design and analy purposes. The notion of well-posedness introduces some typical constraints aimed at insuring their applicability. Definition A standard ODE model ODE(, g) is called well posed if for every signal v(t)E V and for every solution T1: 0, ti- X of(2.4)with 1(0)E Xo there exists a solution R++X of (2. 4)such that c(t)=a1(t)for all t E 0, t1 The OdE from Example 2.1.1 can be used to define a standard autonomous Ode tem model i(t)=-sgn(ar(t)), w(t)=r(t), here V=X=Xo=R, f(a, v, t)=-sgn(a)and g(a, v, t)=r. It can be verified that this autonomous system is well-posed. However, introducing an input into the model destroys well-posedness, as shown in the following example Example 2.2 Consider the standard ODE model i(t)=sgn(ar(t))+o(t),w(t)=r(t) where u(t)is an unconstrained scalar input. Here V=X= XO=R, f(, u, t)=-sgn(r)+v, g(a, v, t) While this model appears to describe a physically plausible situation(velocity dynamics subject to dry friction and external force input u), the model is not well-posed To prove this, consider the input v(t)=0.5= const. It is sufficient to show that no solution of the ODe i(t)=0.5-sgn(x(1) satisfying c(0)=0 exists on a time interval [0, t,] for tf>0. Indeed, let a=r(t) be such solution. As an integral of a bounded function, a =a(t) witll be a continuous function of time. A continuous function over a compact interval always achieves a maximum. Let tm E0, t,] be an argument of the maximum over tE 0, te If c(tm)>0 then tm >0 and, by continuity, r(t)>0 in a neighborhood of tm, hence there exists e>0 such that a(t)>0 for all tE tm-E, tm. According to the differential equation, this means that r(tm-e)=r(tm)+0.5e>a(tm), which contradicts the selection of tm as an argument of maximum. Hence max c(t)=0. Similarly, min c(t)=0. Hence c(t)=0 for all t. But the constant zero function does not satisfy the differentlial equation It can be shown that the absense of solutions in Example 2.1.3 is caused by lack of continuity of function f=f(a, v, t) with respect to r(discontinuity with respect to v and t would not cause as much trouble
3 2.1.3 Well-posedness of standard ODE system models As it was mentioned before, not all ODE models are adequate for design and analysis purposes. The notion of well-posedness introduces some typical constraints aimed at insuring their applicability. Definition A standard ODE model ODE(f, g) is called well posed if for every signal v(t) ⊂ V and for every solution x1 : [0, t1] ∈� X of (2.4) with x1(0) ⊂ X0 there exists a solution x : R+ ∈� X of (2.4) such that x(t) = x1(t) for all t ⊂ [0, t1]. The ODE from Example 2.1.1 can be used to define a standard autonomous ODE system model x˙ (t) = −sgn(x(t)), w(t) = x(t), where V = X = X0 = R, f(x, v, t) = −sgn(x) and g(x, v, t) = x. It can be verified that this autonomous system is well-posed. However, introducing an input into the model destroys well-posedness, as shown in the following example. Example 2.2 Consider the standard ODE model x˙ (t) = −sgn(x(t)) + v(t), w(t) = x(t), (2.6) where v(t) is an unconstrained scalar input. Here V = X = X0 = R, f(x, v, t) = −sgn(x) + v, g(x, v, t) = x. While this model appears to describe a physically plausible situation (velocity dynamics subject to dry friction and external force input v), the model is not well-posed. To prove this, consider the input v(t) = 0.5 = const. It is sufficient to show that no solution of the ODE x˙ (t) = 0.5 − sgn(x(t)) satisfying x(0) = 0 exists on a time interval [0, tf ] for tf > 0. Indeed, let x = x(t) be such solution. As an integral of a bounded function, x = x(t) witll be a continuous function of time. A continuous function over a compact interval always achieves a maximum. Let tm ⊂ [0, tf ] be an argument of the maximum over t ⊂ [0, tf ]. If x(tm) > 0 then tm > 0 and, by continuity, x(t) > 0 in a neighborhood of tm, hence there exists π > 0 such that x(t) > 0 for all t ⊂ [tm − π, tm]. According to the differential equation, this means that x(tm−π) = x(tm)+0.5π > x(tm), which contradicts the selection of tm as an argument of maximum. Hence max x(t) = 0. Similarly, min x(t) = 0. Hence x(t) = 0 for all t. But the constant zero function does not satisfy the differentlial equation. Hence, no solution exists. It can be shown that the absense of solutions in Example 2.1.3 is caused by lack of continuity of function f = f(x, v, t) with respect to x (discontinuity with respect to v and t would not cause as much trouble)
2.2 Existence of solutions for continuous ode This section contains fundamental results establishing existence of solutions of differential equations with a continuous right side 2.2.1 Local existence of solutions for continuous ode In this subsection we study solutions r: [to, t ]bR of the standard ODE i(t=a(a(t),t) (27) (same as(2. 1)), subject to a given initial condition a(to)=? (28) Here a: Z+R is a given continuous function, defined on Z XR. It turns out that a solution =a(t)of (2.7)with initial condition(2. 8)exists, at least on a sufficiently short time interval, whenever the point z0=(o, to) lies, in a certain sense, in the interior Theorem 2.1 Assume that for some r>0 D(x0,to)={(,t)∈R"×R:|-ro≤r,t∈to,to+r]} is a subset of Z. Let M=max{a(z,t川:(z,t)∈D,(x0,to)} Then, for tf= min(to +r/M, to+r] there erists a solution [to, tf]H R of (2. 7) satisfying(2.8). Moreover, any such solution also satisfies I(t)-xo|≤ r for all t∈[o,t Example 2. 3 The ODE i(t)=co+Ci cos(t)+a(t) where co, C1 are given constants, belongs to the class of Riccati equations, which play a prominent role in the linear system theory. According to Theorem 2.1, for any initial condition a(0)=.o there exists a solution of the Riccati equation, defined on some time interval [0, tf] of positive length. This does not however, that the correspond ing autonomous system model (producing output (t)) is well-posed, since such solutions are not necessarily extendable to the complete time half-line 0, oo)
4 2.2 Existence of solutions for continuous ODE This section contains fundamental results establishing existence of solutions of differential equations with a continuous right side. 2.2.1 Local existence of solutions for continuous ODE In this subsection we study solutions x : [t0, tf ] ∈� Rn of the standard ODE x˙ (t) = a(x(t), t) (2.7) (same as (2.1)), subject to a given initial condition x(t0) = x0. (2.8) Here a : Z ∈� Rn is a given continuous function, defined on Z � Rn × R. It turns out that a solution x = x(t) of (2.7) with initial condition (2.8) exists, at least on a sufficiently short time interval, whenever the point z0 = (x0, t0) lies, in a certain sense, in the interior of Z. Theorem 2.1 Assume that for some r > 0 Dr(x0, t0) = {(¯x, t) ⊂ R ¯ n × R : |x − x0| ∀ r, t ⊂ [t0, t0 + r]} is a subset of Z. Let M = max{|a(¯x, t)| : (¯x, t) ⊂ Dr(x0, t0)}. Then, for tf = min{t0 + r/M, t0 + r}, there exists a solution x : [t0, tf ] ∈� Rn of (2.7) satisfying (2.8). Moreover, any such solution also satisfies |x(t) − x0| ∀ r for all t ⊂ [t0, tf ]. Example 2.3 The ODE x˙ (t) = c0 + c1 cos(t) + x(t) 2 , where c0, c1 are given constants, belongs to the class of Riccati equations, which play a prominent role in the linear system theory. According to Theorem 2.1, for any initial condition x(0) = x0 there exists a solution of the Riccati equation, defined on some time interval [0, tf ] of positive length. This does not mean, however, that the corresponding autonomous system model (producing output w(t) = x(t)) is well-posed, since such solutions are not necessarily extendable to the complete time half-line [0,→)
2.2.2 Maximal solutions If 1: to, t1]H+R" and 2: [t1, t2]HR are both solutions of (2.7), and T1(t1)=2(t1), then the function to, t2]++R", defined by (t),t∈[o,t] r2(t),t∈[t1,t2] (i.e. the result of concatenating 1 and x2) is also a solution of(2.7). This means that some solutions of (2.7)can be extended to a larger time interval A solution a: THR of(2.7) is called maximal if there exists no other solution I: THR for which T is a proper subset of T, and i(t)=a(t) for all t E T. In particular, well-posedness of standard OdE system models contains the requirement that ll maximal solutions must be defined on the whole time-line tE 0,oo The following theorem gives a useful characterization of maximal solutions Theorem 2.2 Let X be an open subset of R". Let a: XxRHR be function. Then all maximal solutions of(2.7 are defined on open intervals and, whenever such solution a:(to, t1)HX has a finite interval end t= to E R ort= ti E R(as to=-∞ot1=∞), there erists no sequence tk∈(to,t1) such that th converges to t while a(tk) converges to a limit in X In other words, in the absense of a-priori constraints on the time variable, a solution is not extendable only if a(t)converges to the boundary of the set on which a is defined. In the most typical situation, the domain Z of f in(2. 4) is R"R+, which means no a-priori not extendable over a finite time interval [0,t,),ty< oo, must satisfy the condition e constraints on either r or t. In this case. according to theorem 22. a solution r= lir = In Example 2.2. 1 with co= 1, C1=0, one maximal ODE solution is a(t)= tan(t) defined for tE(T/ 2, / 2). It cannot be extended on either side because x(t)l-+oo as t→丌/2ort→-丌/2 2.2.3 Discontinuous dependence on time The ode describing systems dynamics are frequently discontinuous with respect to the time variable. Indeed, the standard Ode system model includes i(t)=f(r(t), u(t),t) where v=v(t) is an input, and the Ode becomes discontinuous with respect to t when ever v is a rectangular impulse etc. As long as the time instances at which a(a, t)
5 2.2.2 Maximal solutions If x1 : [t0, t1] ∈� Rn and x2 : [t1, t2] ∈� Rn are both solutions of (2.7), and x1(t1) = x2(t1), then the function x : [t0, t2] ∈� Rn, defined by ⎩ x1(t), t ⊂ [t0, t1], x(t) = x2(t), t ⊂ [t1, t2], (i.e. the result of concatenating x1 and x2) is also a solution of (2.7). This means that some solutions of (2.7) can be extended to a larger time interval. A solution x : T ∈� Rn of (2.7) is called maximal if there exists no other solution ¯ x¯ : T ¯ ¯ ∈� Rn for which T is a proper subset of T, and x(t) = x(t) for all t ⊂ T. In particular, well-posedness of standard ODE system models contains the requirement that all maximal solutions must be defined on the whole time-line t ⊂ [0,→). The following theorem gives a useful characterization of maximal solutions. Theorem 2.2 Let X be an open subset of Rn. Let a : X × R ∈� Rn be a continuous function. Then all maximal solutions of (2.7) are defined on open intervals and, whenever such solution x : (t ¯ 0, t1) ∈� X has a finite interval end t = t0 ⊂ R or t ¯ = t1 ⊂ R (as opposed to t0 = −→ or t1 = →), there exists no sequence tk ⊂ (t0, t1) such that tk converges to t ¯ while x(tk) converges to a limit in X. In other words, in the absense of a-priori constraints on the time variable, a solution is not extendable only if x(t) converges to the boundary of the set on which a is defined. In the most typical situation, the domain Z of f in (2.4) is Rn×R+, which means no a-priori constraints on either x or t. In this case, according to Theorem 2.2, a solution x = x(t) not extendable over a finite time interval [0, tf ), tf < →, must satisfy the condition lim |x(t)| = →. t�tf In Example 2.2.1 with c0 = 1, c1 = 0, one maximal ODE solution is x(t) = tan(t), defined for t ⊂ (−�/2, �/2). It cannot be extended on either side because |x(t)| � → as t � �/2 or t � −�/2. 2.2.3 Discontinuous dependence on time The ODE describing systems dynamics are frequently discontinuous with respect to the time variable. Indeed, the standard ODE system model includes x˙ (t) = f(x(t), v(t), t), where v = v(t) is an input, and the ODE becomes discontinuous with respect to t whenever v is a rectangular impulse etc. As long as the time instances at which a(x, t) is
discontinuous for a fixed finite set ti t20 (a) the set D(x0,to)={(x,t)∈R"×R:|-ro≤r,t∈to,to+r] is a subset of z (b) the function tH a((t), t)is integrable on [to, to +rl for every continuous function r:[to,to+r]→ R" satisfying az(t)-ro|≤ r for all t∈[to,to+r] (c) for every e>0 there erists8>0 such that a(x1(t),t)-a(x2(1,t)t0 is integrable over every finite interval, and the inequality ( -x1(t)-t-3r2(t)ldt0 t=0 does not have a solution on [0, oo)for every To#0. Indeed, if 0, ti]HR is a solution t1>0 then 0 for all t+0. Hence a(t)=ct for some constant c, and x(0)=0
� � � 6 discontinuous for a fixed finite set t1 0 (a) the set Dr(x0, t0) = {(x, t) ⊂ Rn × R : |x − x0| ∀ r, t ⊂ [t0, t0 + r]} is a subset of Z; (b) the function t ∈� a(x(t), t) is integrable on [t0, t0 + r] for every continuous function x : [t0, t0 + r] ∈� Rn satisfying |x(t) − x0| ∀ r for all t ⊂ [t0, t0 + r]; (c) for every π > 0 there exists � > 0 such that t0+r |a(x1(t), t) − a(x2(t), t)|dt 0 x˙ (t) = x(0) = x0 0, t = 0, does have a solution on [0,→) for every x0 ⊂ R (in this particular case the solutions can be found analytically). Indeed, for every continuous function x : [0,→) ∈� R the function t ∈� t −1/3x(t) for t > 0 is integrable over every finite interval, and the inequality t1 t1 |t −1/3 x1(t) − t −1/3 x2(t)|dt ∀ t −1/3 dt max |x1(t) − x2(t)| 0 0 t�[0,t1] holds. On the contrary, the differential equation ⎩ t −1x(t), t > 0 x˙ (t) = x(0) = x0 0, t = 0, does not have a solution on [0,→) for every x0 ∞= 0. Indeed, if x : [0, t1] ∈� R is a solution for some t1 > 0 then ⎧ ⎨ d x(t) = 0 dt t for all t ∞= 0. Hence x(t) = ct for some constant c, and x(0) = 0
2. 2. 4 Differential inclusions Let X be a subset of R", and let n: X-2 be a function which maps every point of X to a subset of r". Such a function defines a differential inclusion i(t)∈m(x(1) (29) By a solution of (2. 1) on a convex subset T of R we mean a function a: TH X suck )=/ u(tyt,t2∈T for some integrable function u: THR satisfying the inclusion u(t)E n(a(t) for all t E T. It turns out that differential inclusions are a convenient, though not always adequate, way of re-defining discontinuous ode to guarantee existence of solutions It turns out that differential inclusion(2. 9 )subject to fixed initial condition r(to)=.o has a solution on a sufficiently small interval T=to, ti whenever the set-valued function n is compact conver set-valued and semicontinuous with respect to its argument(plus, as usually, o must be an interior point of X) Theorem 2.4 Assume that for some r>0 (a) the set B(x0)={x∈R":|x-ro≤r} is a subset of X (b) for every i E Br (ro) the set n(i) is conver (c) for every sequence of Tk E B,(ro) converging to a limiti E B,(ro) and for every sequence uk∈m(k) there erists a subsequence k=k(q)→∞asq→ oo such that the subsequence ik(g has a limit in n(a) Then the supremum M=sup{|叫:n∈n(),∈Dn(xo,to)} finite, and tf=minto+r/M, to +r] there erists a solution a: [to, t H R of (2.9)satisfying a(to)=o. Moreover, any such solution also satisfies lx(t)sol s r for all tE [to, tf] The discontinuous differential equation i(t)=-sgn(a(t))+c
7 2.2.4 Differential inclusions Let X be a subset of Rn, and let � : X � 2Rn be a function which maps every point of X to a subset of Rn. Such a function defines a differential inclusion x˙ (t) ⊂ �(x(t)). (2.9) By a solution of (2.1) on a convex subset T of R we mean a function x : T ∈� X such that � t2 x(t2) − x(t1) = u(t)dt � t1, t2 ⊂ T t1 for some integrable function u : T ∈� Rn satisfying the inclusion u(t) ⊂ �(x(t)) for all t ⊂ T. It turns out that differential inclusions are a convenient, though not always adequate, way of re-defining discontinuous ODE to guarantee existence of solutions. It turns out that differential inclusion (2.9) subject to fixed initial condition x(t0) = x0 has a solution on a sufficiently small interval T = [t0, t1] whenever the set-valued function � is compact convex set-valued and semicontinuous with respect to its argument (plus, as usually, x0 must be an interior point of X). Theorem 2.4 Assume that for some r > 0 (a) the set Br(x0) = {x ⊂ Rn : |x − x0| ∀ r} is a subset of X; (b) for every x¯ ⊂ Br(x0) the set �(¯x) is convex; (c) for every sequence of x¯k ⊂ Br(x0) converging to a limit x¯ ⊂ Br(x0) and for every sequence u¯k ⊂ �(¯xk) there exists a subsequence k = k(q) � → as q � → such that the subsequence u¯k x). (q) has a limit in �(¯ Then the supremum M = sup{|u¯ ¯ | : u ⊂ �(¯ ¯ x), x ⊂ Dr(x0, t0)} is finite, and, for tf = min{t0 + r/M, t0 + r}, there exists a solution x : [t0, tf ] ∈� Rn of (2.9) satisfying x(t0) = x0. Moreover, any such solution also satisfies |x(t) − x0| ∀ r for all t ⊂ [t0, tf ]. The discontinuous differential equation x˙ (t) = −sgn(x(t)) + c
here c is a fixed constant, can be re-defined as a continuous differential inclusion(2.9 by introducing y>0, n(y)=e-1,c+1,y=0 y<0 The newly obtained differential inclusion has the "existence of solutions"'property, and appears to be compatible with the dry friction"interpretation of the sign nonlinearit In particular, with the initial condition a(0)=0, the equation has solutions for every value of c E R. If cE[-1, 1, the unique maximal solution is a(t)=0, which corresponds to the friction force"adapting" itself to equalize the external force, as long as it is not too large The differential inclusion model is not as compatible with the "on/off controller interpretation of the sign nonlinearity. In this case, due to the unmodeled feedback loop delays, one expects some "chattering"solutions oscillating rapidly around the point To=0. It is possible to say that, in this particular case, the solutions of(2.9 describe the limit behavior of the closed loop solutions as the loop delay approaches zero
� � 8 where c is a fixed constant, can be re-defined as a continuous differential inclusion (2.9) by introducing � {c − 1}, y > 0, �(y) = [c − 1, c + 1], y = 0, {c + 1}, y < 0. The newly obtained differential inclusion has the “existence of solutions” property, and appears to be compatible with the “dry friction” interpretation of the sign nonlinearity. In particular, with the initial condition x(0) = 0, the equation has solutions for every value of c ⊂ R. If c ⊂ [−1, 1], the unique maximal solution is x(t) ≤ 0, which corresponds to the friction force “adapting” itself to equalize the external force, as long as it is not too large. The differential inclusion model is not as compatible with the “on/off controller” interpretation of the sign nonlinearity. In this case, due to the unmodeled feedback loop delays, one expects some “chattering” solutions oscillating rapidly around the point x0 = 0. It is possible to say that, in this particular case, the solutions of (2.9) describe the limit behavior of the closed loop solutions as the loop delay approaches zero
