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!