Problem 3.1. Consider a system described by the elgebraic input-output description: y(t)=mu(t)+b Find the conditions that m and b must satisfy such that the system is linear Problem 3.2. Chook if a system defined by the following input-outpet description is linear or not: dtdy(t)+2ty(t)=2u(t) Problem 3.3. Consider a system haring fro inputs and two eutpuits. Assume that the system is algebraic and linear. If we know that y(t)=[21]T=T([11]T)y(t)=[34]T=T([12]T) then determine the system response when the input excifation is u(t) =[11]T Problem 3.4. Consider an eloctronic amplifier with input voltage n2(t) and output soltage vv(t). Assume that v0(t)=8vv(t)+2 3.4.1 Show that the amplifier docs not strielly satisfy the principle of superposition. Thas this system is not strictly linear. (A better term for this system mould be affine.) 3.4.2 Note that the system ann also be written: t0(t)=8v4(t)+2d4(t) where d1(t) is a constant offset (equal to 1 ). Show that the principle of superposition does hold for the input vector [v(t)d4(t)]2 Problem 3.5. Consider an electronic amplifier, similar to that in Problem 3.4 but having input voltage v(t) and output noltage v0(t) related by v0(t)=8(v1(t))2 3.5.1 Show that this amplifier is nonlinear, by showing that if we compose v4(t) from two swb-components, i.e v1(t)=va(t)+vv2(t), then the response to v1(t) is not the sum of the responses to va(t) and val (t). 3.5.2 Assume that v1(t)=5+cos(100t). What is the outputy. Figure 3.7. Dynamic mechanical system Problem 3.1. Consider a system described by the elgebraic input-output description: y(t)=mu(t)+b Find the conditions that m and b must satisfy such that the system is linear Problem 3.2. Chook if a system defined by the following input-outpet description is linear or not: dtdy(t)+2ty(t)=2u(t) Problem 3.3. Consider a system haring fro inputs and two eutpuits. Assume that the system is algebraic and linear. If we know that y(t)=[21]T=T([11]T)y(t)=[34]T=T([12]T) then determine the system response when the input excifation is u(t) =[11]T Problem 3.4. Consider an eloctronic amplifier with input voltage n2(t) and output soltage vv(t). Assume that v0(t)=8vv(t)+2 3.4.1 Show that the amplifier docs not strielly satisfy the principle of superposition. Thas this system is not strictly linear. (A better term for this system mould be affine.) 3.4.2 Note that the system ann also be written: t0(t)=8v4(t)+2d4(t) where d1(t) is a constant offset (equal to 1 ). Show that the principle of superposition does hold for the input vector [v(t)d4(t)]2 Problem 3.5. Consider an electronic amplifier, similar to that in Problem 3.4 but having input voltage v(t) and output noltage v0(t) related by v0(t)=8(v1(t))2 3.5.1 Show that this amplifier is nonlinear, by showing that if we compose v4(t) from two swb-components, i.e v1(t)=va(t)+vv2(t), then the response to v1(t) is not the sum of the responses to va(t) and val (t). 3.5.2 Assume that v1(t)=5+cos(100t). What is the outputy. Figure 3.7. Dynamic mechanical system
