Question

Problem 1 (25 points): Consider a system described by the differential equation: +0)-at)y(t) = 3ú(1); where y) is the system output, u) is the system input, and a(t)is a function of time t. o) (10 points): Is the system linear? Why? P(15 points): Ifa(t) 2, find the state space equations?

Answer 1

Given differential equation is ý (t) + ý ct - Q(+) y (at) = 3.4(+) act where is a function of time it'. y(t) is the system output u(t) is the system Input and remaining Here t is a independent variable i are dependent variables, In the given equation, are of o The dependent variable and its derivatives first degree only, o There is no product of dependent variables and for its derivatives. pependent variables and its o derivatives are not arguments of any non linear equations or functions. equation is binear.. Hence the given differential Then Given alt=21 the differential equation becomes 4 + (4 - 8) 9 (8) = (3) (4)

It can be written as Apply Laplace transform s?. y(s) + 5.4(s) - 4.4() = 3.40). S 4039[5'75 - 23 = u() [35] YOO KO = (35) Gaas-a) s equation o that they se SixS) + Six(S)-.x) = V(S) Replacing 'sa by nema Det - 2.x = u(t) Let do = * = x= x sq + xq - 209a USA a q = px -+ uct)

☺ Lille w JL x) y equation o A=621] , B=(9) { Facm = Ax+ Bu standard equations And we co X (S) Y(S) = 3.5.x(s) Apply Inverse la place y (19 = 39 (2 cm) y(t) = (3). £, = (3)683) y(t) = 3.42 ys. [o 3) (3.3 is equation ® Į. From y = cxto. XHOW? D= [0] Here, for the given differential equation in and @ equations © are state space question ; equations-