Linear-quadratic Optimal Controllers and the Kalman Filter

FINAL PROJECT
PROJECT NO. 3: FINAL PROJECT
LINEAR-QUADRATIC OPTIMAL CONTROLLERS AND THE KALMAN
FILTER
For this project, an LTI (linear time invariant), third order SISO (Single input single output) was
selected [1]. The system is completely controllable and observable. The system state space
representation is as shown below.
[ ̇x1
̇x2
̇x3 ] = [ 0 1 0
0 0 1
−8 −12 −5 ] [ x1
x2
x3 ] + [ 0
0
6 ] [ u ]
[ y ]= [ 1 0 0 ] [ x1
x2
x3 ]
To check for controllability and observability of the system the MATLAB code in part 1 of the
appendix was ran. It was observed that the system was completely controllable and observable
and therefore system state space met the requirements.
Part 1a: Zero Set Point Linear-Quadratic Controller
It was assumed that all state space variables were available for feedback. The optimal
performance criterion weighted matrices were given by:
R1=CT C= [1 0 0
0 0 0
0 0 0 ]
R2=I m=I 3=1
The optimal feedback gain, Fopt, was obtained by using the MATLAB function lqr ([Fopt,
P]=lqr(A, B, R1,R2]). The assumptions made was that all components of the initial condition
state space vector were equal to 1. Fopt was obtained as:
Fopt=[0.3333 0.1585 0.0311]
The optimal value of the performance criterion of the system was obtained using the line,
Jopt=0.5*X0’*P*X0. Jopt was obtained as:
Jopt =0.9516
The mathematical model of the zero-set points linear-quadratic controller designed was then
simulated in MATLAB Simulink. Figure 1 below shows the block diagram in Simulink.

FINAL PROJECT
Figure 1: Block of the controller in Simulink
The following plots were made by running the simulation.
Figure 2: Simulation results for input and output signal

FINAL PROJECT
Figure 3: Variation of the state variables with time
Part 1b: Non-zero Set Point Optimal Linear-Quadratic Controller
For this design was selected as:
Cz =[1 0 0]
The desired steady sate value for the system-controlled output: was selected as 1.
The optimal linear-quadratic non-zero set point controller was then designed with the weighted
matrices and the initial condition defined in Part 1a using the code in the appendix. The system-
controlled output was then plotted.
Figure 4: Simulink block diagram of the designed controller.