Question (a)
From the second equation, we get
Substituting this equation in the first one, we get
Let us take the state variables as
Let the inputs be
From the above definition, we get one state space equation as
So
Now substituting the state variables in the differential equation we get
Substituting the state variables in , we get
So the state equations are
The output equations are
Question (b)
The state space representation in matrix form is
So the matrices are
Question (c)
The state space diagram is as shown below
Transfer function is given by
So the inverse
So the transfer function will be
So the transfer function matrix is
Question 5 Following differential equations defines input-output relationships of a system with y as output and...
Problem 2 - System Representation: Input/Output (20pts) (2 For the following set of coupled differential equations: or the following set dx dt + F(t dt Find the input/output equation describing x, given the input force F(t). Problem 2 - System Representation: Input/Output (20pts) (2 For the following set of coupled differential equations: or the following set dx dt + F(t dt Find the input/output equation describing x, given the input force F(t).
5. A two-input, two-output dynamic system is defined by the following differential equations system 2x, (t) ) +3x1(t) -2x2 () fi(t) x2 t) -2x,(t) +2x2(t) f2(t) Determine its transfer function matrix considering that the input is (fi (t) f2(t) and the output is x, (t) x2 (t)J. 5. A two-input, two-output dynamic system is defined by the following differential equations system 2x, (t) ) +3x1(t) -2x2 () fi(t) x2 t) -2x,(t) +2x2(t) f2(t) Determine its transfer function matrix considering that...
Question 3 Consider the following linear system of differential equations dx: = 2x-3y dt dy dt (a) Write this system of differential equations in matrix form (b) Find the general solution of the system (c) Solve the initial value problem given (0) 3 and y(0)-4 (d) Verify the calculations with MATLAB Question 3 Consider the following linear system of differential equations dx: = 2x-3y dt dy dt (a) Write this system of differential equations in matrix form (b) Find the...
Determine a set of state equations and an output equation for the system that is described by the following differential equation. Put the results in matrix form y''(t)+7Y'(t)+3y(t)=4u'(t)+5u(t)
Consider three systems with the following input-output relationships 6. Consider three systems with the following input-output relationships: { 4 0, odd System 1: y[n n even r[n] 10ar(n 2]3r[n - 1 System 2: yn + + System 3: yn x[3n] The interconnection diagram is at follows: System 1 System 2 System 3 Find the input-output relationship of the interconnected system. State the properties of the system (linear, stable, time invariant, memoryless, and causal). 6. Consider three systems with the following...
Consider the following linear system of differential equations: dx/dt = 2x-3y dy/dt = -x +4y (a) Write this system of differential equations in matrix form (b) Find the general solution of the system (c) Solve the initial value problem given x(0) = 3 and y(0) = 4 (d) Verify the calculations with MATLAB
3. (l’+2° +1²=4') Topic: Laplace transform, CT system described by differential equations, LTI system properties. Consider a differential equation system for which the input x(t) and output y(t) are related by the differential equation d’y(t) dy(t) -6y(t) = 5x(t). dt dt Assume that the system is initially at rest. a) Determine the transfer function. b) Specify the ROC of H(s) and justify it. c) Determine the system impulse response h(t).
Given the system of differential equations o y (7tcos(tut) Write the first order matrix differential equation that is the basis for using Euler's method to compute the numerical solution. It is assumed you will use two auxiliary functions, xi and t2 Define the functions i and 2 in terms of v and y. E2 dri (t) dt 1(t) dr2(t) dt a2(t) Given the system of differential equations o y (7tcos(tut) Write the first order matrix differential equation that is the...
For the given RC circuit shown below, ys the output, and ut) is the input. Values of the components are marked on schematic i) Derive the system differential equation and transfer function Y(s)/U(s) ii) Choose voltage across capacitors as states and derive the state equations and state matrices (A, B, C,and D). iii) Validate the states by deriving the transfer function from state matrices. iv) Choose a different set of states and derive a different state equation and state Matrix...
DONGGUK UNIVERSITY Division of Electronics and Electrical Engineering ENE4067-01 Modern control system Homework #2 Student ID: 1st Semester, 2020 Name: Note that you have to represent all the derivation process in detail. Problem 1. A spring-mass-friction system is described by the following differential equation. d?y(t) dy(t) dt2 + y(t) = r(t) dt (a) Define the state variables as xi(t) = y(t), x2(t) = y(t) + dy(t)/dt. Write the state equations in vector-matrix form. Find the state transition matrix of A....