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1. In the following two block diagrams, all parameters A, B and K are positive real...
Applied Differential Equations 2. For which parameters a and b the zero solution of the following systems is asymp- totically stable? ¿ = x +ay+ya, y = bx – 3y – x2 = y + sin x, y = ax + by.
Q.3(a) Transfer function model of a plant is, G(s) The controller is Ge(s)-K, where K is a constant. Find the value of K such that steady-state error for unit ramp input is 0.1. Find the gain margin and the phase mar gin (6 marks) (b) What are the effects on gain margin, phase margin and steady-state error, if the gain K is increased? (3 marks (c) Can the closed loop be unstable if the controller of Q.3(a) is implemented digi...
Let a be a positive real number. Consider a discrete-time echo system (called system 1) given by the difference equation y[n] = v[n] + av[n – 4). Here v is the input signal and y is the output signal. A. (1 mark) Determine the systems' transfer function H1 (2). B. (1 mark) What are the pole(s) of this system? Plot the pole(s) in the complex plane. C. (2 marks) Is this system stable? Explain your answer. D. (2 marks) Determine...
Find the most general real-valued solution to the linear system of differential equations (1 point) a. Find the most general real-valued solution to the linear system of differential -5 -36 x. -5 equations x 1 CHH x1 (t) = C1 x2 (t) b. In the phase plane, this system is best described as a O source/ unstable node Osink /stable node Osaddle center point ellipses Ospiral source spiral sink none of these tsi O O O (1 point) a. Find...
The systems represented by these block diagrams can each be described by a differential equation of the form IN (y(t))-aN-1dNī(w(t)).t +a2 g@) +«Яу(t)) +aoy(t) rće For each system, what is the value of N? For each system, what are the a coefficients, starting with aNand going down to N-1 ao? In system (b), what range of values of A will make the system stable? The value of A for the given system is 3. -3 -2 y(t) The value of...
. Question 1 (40 marks) This question asks you to demonstrate your understanding of the following learning objectives LO 1.6 Express the Laplace Transform of common mathematical functions and linear ordinary differential equations using both first principles and mathematical tables. LO 1.7 Construct transfer functions for linear dynamic systems from (i) differential equations and (ii) reduction of block diagrams. LO1.8 Determine the time response of a Linear SISO system to an arbitrary input and having arbitrary initial conditions. LO 1.9...
The state variable model of the two tanks process is given by the equations r1 10 01 r1o 2 0-1 lu Tank 1 Tank 2 Explain the differential equations for the tanks Draw the block diagram for the system model * .Modify the block diagram to realize the system model by first order transfer functions: 1+Ts Determine the controllability and observability of the system model Design a full-state feedback with the eigen values λ-λ2--2 of the closed loop system Design...
(d)A first-order system is described by the following differential equation +24x(t) = r(1) dī i) Discretize the system using (a) forward (Euler) approximation, and (b) backward (Euler) approximation, respectively, with a sampling time of T - 0.1 for both. Write down the two resulting systems as difference equations. 20% (ii) Check the stability of the continuous-time system described by equation (2) 10% (iii) Discuss the stability of the above two discretised systems obtained in (d(i)). Explain how to choose the...
Consider the linear system given by the following differential equation y(4) + 3y(3) + 2y + 3y + 2y = ů – u where u = r(t) is the input and y is the output. Do not use MATLAB! a) Find the transfer function of the system (assume zero initial conditions)? b) Is this system stable? Show your work to justify your claim. Note: y(4) is the fourth derivative of y. Hint: Use the Routh-Hurwitz stability criterion! c) Write the...
a. d. Match the following z-plane diagrams to their corresponding difference equations 1.5 1.5 0.5 .' 05 05 05 Real Part Real Pat 1.5 1.5 0.5 0.5 05 05 ' Real Part Real Part 1.5 1.5 0.5 05 Real Part Real Part Difference Equation Options: Equation 1: y[n] 0.5yIn-1] + xm Equation 2: yn0.9yfn - 1] + 0.5x[n +0.5xn -1] Equation 3: y[n] =-0.9y[n-1] + 0.5z[n] _ 0.5z[n-1] Equation 4: y[n] = z[n] + x[n-1] + z[n-2 + z[n-3] Equation...