I MIUIIZUIU LUIStall. 5.9 Consider da 30 = [ + ]00+[ 6 ]use i(t) = y(t)=[-2...
Using Differential
Equations.
6. For y, = y3 _ y, y(0) = 30, -00 <30 < 00, draw the graph of (y) = y3-y versus y, determine the equilibrium solutions (critical points) and classify each one as unstable or asymptotically stable. Draw the phase line, and sketch several representative integral curves (graphs of solutions) in the (t, y) plane. Hint: None of this requires explicit formulas for solutions y = φ(t) of the initial value problem.]
2. (Chapter 2). A linear, time-invariant, continuous-time (LTIC) system with input f(t) and output y(t) is specified by the differential equation D2(D +1)y(t) (D - 3)f(t) Find the characteristic polynomial, characteristic equation, characteristic root(s), and characteristic mode(s) of this system. a. b. Is this system asymptotically stable, marginally stable, or unstable? Justify your answer.
2. (Chapter 2). A linear, time-invariant, continuous-time (LTIC) system with input f(t) and output y(t) is specified by the differential equation D2(D +1)y(t) (D - 3)f(t)...
Problem #4 [15 Points] Given -1001 00-1」 11 0 1 00 (a) Find the impulse response h(t) and the transfer function matrix H(s). Which modes are present in h(t)? What are the poles of H(s)? b) Is the system asymptotically stable? Is it BIBO stable? Explain. (c) ls it possible to determine a linear state feedback control law u = Fr+r so that all closed loop eigenvalues are at -1? If yes, find such F
2. (Chapter 2). A linear, time-invariant, continuous-time (LTIC) system with input 1(1) and output y(t) is specified by the differential equation D(D? + 1)y(t) = Df(t). a. Find the characteristic polynomial, characteristic equation, characteristic root(s), and characteristic mode(s) of this system. b. Is this system asymptotically stable, marginally stable, or unstable? Justify your answer.
(30%) Consider a system with the transfer function Y(s) s+6-k (a) Determine the range of parameter k so that the system G(s) is stable. (b) Determine the value of k for which the system becomes marginally stable. (c) Assuming parameter k has the value in part(b) and hence the system is marginally stable, find a bounded input r(t) that results in unbounded output y(t). For this part, specifying the bounded input signal r(t) and a justification is enough Finding v(t)...
B is a rectangular region and its coordinates are (0,0),(5,0),6)3. Please compute S[(x + y)da with using x=2u-3V.X=2u+3V. this integral I = I B
2) An LTI DT system is defined by the difference equation: y[n] = -0.4yIn - 1] + x[n]. a) Derive the impulse response of the system. (2 pt) b) Determine if the system is BIBO stable. (1 pt) c) Assuming initial conditions yl-1) = 1, derive the complete system response to an input x[n] = u[n] - u[n-2), for n > 0.(2 pt) d) Derive the zero-state system response to an input z[n] = u[n] - 2u[n - 2] +...
5. Consider the system given in (a) is marginally stable. X + 4. 10/( s (0.1 s +1) 1/s G(s) (a) Find G(s) (b) Determine Y(s)/X(s) in terms of G(s). (c) If the error E(s)-X(s)-Y (s) determine E(s)/Y(s). (d) Determine the steady-state value of e(t) given that s(t): u(t) 6. Consider the system given in (a) is marginally stable. X+ G(s) (a) Determine the transfer function (s)/X(s). (b) If the error e()-x(0)-y() determine a G(s) such that e(oo) -1/2 when...
Using Change of Variables..Evaluate ∫∫ R 15y/x dA where R is the region bounded by xy = 2, xy = 6 , y = 4 and y =10 usingthe transformation x=v , y=2u/3v.
1. For the differential equation (y-y-6) șin(y/2) a) Find the critical points for y in (-6,6) and lassify the critical points as asymptotically stable, or unstable, or semi stable. b) Sketch approximate but clear solutions corresponding to the initial conditions 1.0 -0.8 -0.6 -0.4 0.2 0.2 0.4 0.6 0.8 1.0 -2 .6
1. For the differential equation (y-y-6) șin(y/2) a) Find the critical points for y in (-6,6) and lassify the critical points as asymptotically stable, or unstable, or semi...