Determine whether the system is observable

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Reference no: EM132898076

SEE701 Control Systems Engineering - Deakin University

QUESTION 1

Consider the following state-space model

x?^·(t) = Ax(t) + Bu(t)
Y(t) = Cx(t) + Du(t)

where matrices A, B, C, D are as given below:

A=[-3 -2 -0.75 0 0; -3 0 0 0 0; 0 2 0 0 0; 0 0 1 0 0; 0 0 0 2 0];

B=[1;0;0;0;0];

C=[0 0 0 0 2.75];

D=0.
Determine whether this system is observable.

QUESTION 2

Consider the following state-space model

x?^·(t) = Ax(t) + Bu(t)
Y(t) = Cx(t) + Du(t)

Where matrices A, B, C, D are as given below:

A=[-1 1; 1 -2]; B=[1;2]; C=[0 1]; D=0.

(a) Determine whether the system is observable.

(b) Design a full-order state observer to estimate the state vector of the system. In your observer stet design, pick the poles of the state observer at {-a, -0.5b}, where a and b are obtained from your student ID number.

(c) Use Matlab to simulate and show that the designed state observer correcltly estimates the true state variables of the system for any initial condition. Include all codes and figures.

(d) Design a reduced-order state observer for the system following the five steps (as outlined in the lecture notes, chapter 7) and pick the observer's pole at -a.

(e) Use Matlab to simulate and show that the designed reduced-order state observer correcltly estimates the state variable x1(t) of the system for any initial condition. Include all codes and figures.

(f) Now that you have done the design for both state observers. Take a moment to reflect and explain the differences between them. Please list down some possible advantages of the reduced-order observer over the full-order state observer.

QUESTION 3

Consider a system described by the following differential equation:

y¨ - 2y^· + y = 3u¨ - 2u

(a) Derive a state-space representation of the system.

(b) Check the stability, controllability and observability of the system.

(c) Design a suitable PI controller for the system so that the output tracks a step reference input. Use Matlab to simulate and plot the response of the closed-loop system to a unit-step reference input. Does the output now track the reference unit-step input? Show your full working including all codes and plot from Matlab.

(d) Design a suitable full-order state observer to estimate the state vector of the system. Use Matlab to simulate and show that the designed state observer correcltly estimates the true state vector of the system for any initial condition. Include all codes and figures.

(e) Combined both the state observer and the PI controller and use Matlab to simulate and plot the response of the closed-loop system to a unit-step reference input. Show your full working including all codes and plot from Matlab.

Question 4

Consider a system described by the following differential equation:

y¨+ 2yy¨+ 3y· + 4y = 3u¨ - 2u

(a) Derive a state-space representation of the system.

(b) Check the stability and observability of the system.

(c) Design a reduced-order state observer for the system following the five steps (as outlined in the lecture notes, chapter 7) and pick the observer's pole at {-a, -0.5b}.

(d) Use Matlab to simulate and show that the designed reduced-order state observer correcltly estimates the state vector of the system for any initial condition. Include all codes and figures.

Attachment:- Control Systems Engineering.rar

Reference no: EM132898076

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