Simulation & Optimization of Linear Systems
Budget: $30 – $250 USD
Eigenvalue Assignment and LQ Optimal Controllers
Eigenvalue Assignment Non-Zero Set-Point Controlled Design
a) 5 pts. Take any linear system of your interest that is controllable. There are several such systems in
the books [1]-[4]. Select an arbitrarily set of asymptotically stable eigenvalues and simulate the
corresponding system response with full-state feedback. All state variables are supposed to go to zero as
time increases, and hence the system output. Set all system initial conditions to 1. Plot the corresponding
system response. Choose a desired value for the system output and design the eigenvalue assignment non-
zero set controller with the same set of desired closed-loop eigenvalues. Plot the system output that settles
down at its desired value.
[1] A. Sinha, Linear Systems: Optimal and Robust Control, Taylor& Francis, 2007 (our textbook).
[2] B. Anderson and J. Moore, Optimal Control Linear-Quadratic Methods (posted on Canvas).
[3] Z. Gajic and . Lelic, Modern Control Systems Engineering, Prentice Hall, 1996 (posted on Canvas).
[4] R. Beard, with T. McLain, C. Peterson, M. Killpack, Introduction to Feedback Control: Using
Design Studies, 2023.
Hint 1: Before you start writing the project report read the “required format for the project report” given
at the end of the project assignment. Describe in sufficient detail any technique used, including the
corresponding formulas.
Hint 2: If in this part you have a MIMO system, then you need to use the block diagram and derivations
from Lecture 8a, pages 15-16. Please note that feedback is from the state variables ( )x t not from the
controlled output ( )z t r= . If you have a SISO system then you can use a non-zero tracking result formula
from Lecture 5b (based on Section 3.1.3 from the textbook) , page 9, and the corresponding block diagram.
Design of Optimal Linear-Quadratic (LQ) Controllers
b) 5 pts. Non-Zero Set Point Optimal LQ Controller. Consider the system from part a) or take any
other linear controllable system that you are interested in. Choose the weighed (penalty) matrices such that
the positive definite stabilizing solution of the algebraic Riccati equation exists (see Lecture 6a; textbook
Section 3.5.4). Find the optimal feedback gain and optimal performance values when all initial conditions
are equal to 1. Choose the desired non-zero value for the system output. Follow presentation from Lecture
8a, pages 15-17, and design the Simulink block diagram for the corresponding controller. Plot the obtained
optimal state trajectories that show that the system response settles down at the desired value.
c) 5 pts. Optimal LQ Controller with the Prescribed Degree of Stability. Consider the system from
Part b) or take any other linear controllable system. Based on information obtained in Part b) regarding the
optimal closed-loop optimal eigenvalues, design the optimal LQ controller with the prescribed degree of
stability (Lecture 6b) that moves all closed-loop eigenvalues by two units to the left (deeper into the stability
region,
α = >2 0 ). Plot the obtained optimal closed-loop system response and compare it to the response
obtained in Part b). Observe the impact of the eigenvalue shift on the transient response and the feedback
gain. Assume that all initial conditions are equal to 1. Find the optimal performance criterion.
d) 5 pts. Non-Set Point Optimal LQ Controller with Constant Disturbance. Assume that the system
in Part b) is subjected to a constant input disturbance. An integrator in the feedback loop will be needed to
reject the impact of the constant disturbance so that the desired non-zero output signal can maintain the
same value as in Part b). Design the corresponding LQ optimal controller by following presentation in
Lectures 7b and 8b. Build the corresponding SIMULINK block diagram and plot the results obtained.
Compare the obtained output signal with the system output from Part b).
Hint: See the general detailed derivations of this technique in Lecture 8c (not presented in class, it has
just been posted), where optimal control has (in addition to a constant feedforward term) two feedback
terms: one from the system state variables and another one from the integral of the system output error
( ) ( )desiredy t y t− .
Comment: This problem could have been solved using the presentation of Lecture 7a (Section 3.7 from
the textbook), in which case optimal control will have only one feedback from the system state variables,
in addition to a constant feedforward term.
Required Format for the Project Report
The report should contain the following parts:
1) Project formulation.
2) Provide a brief introduction of the techniques used, including all relevant math formulas.
3) Plot all signals and figures as required by the project. Plot the figures from the MATLAB window
and not from SIMULINK scopes.
4) Include block diagrams (generated by Simulink or any other software) in the main text.
5) Provide comments on the obtained results.
6) Write conclusions.
7) Provide a list of references.
8) Appendix: MATLAB code, and anything else that you consider relevant to the project.
Eigenvalue Assignment Non-Zero Set-Point Controlled Design
a) 5 pts. Take any linear system of your interest that is controllable. There are several such systems in
the books [1]-[4]. Select an arbitrarily set of asymptotically stable eigenvalues and simulate the
corresponding system response with full-state feedback. All state variables are supposed to go to zero as
time increases, and hence the system output. Set all system initial conditions to 1. Plot the corresponding
system response. Choose a desired value for the system output and design the eigenvalue assignment non-
zero set controller with the same set of desired closed-loop eigenvalues. Plot the system output that settles
down at its desired value.
[1] A. Sinha, Linear Systems: Optimal and Robust Control, Taylor& Francis, 2007 (our textbook).
[2] B. Anderson and J. Moore, Optimal Control Linear-Quadratic Methods (posted on Canvas).
[3] Z. Gajic and . Lelic, Modern Control Systems Engineering, Prentice Hall, 1996 (posted on Canvas).
[4] R. Beard, with T. McLain, C. Peterson, M. Killpack, Introduction to Feedback Control: Using
Design Studies, 2023.
Hint 1: Before you start writing the project report read the “required format for the project report” given
at the end of the project assignment. Describe in sufficient detail any technique used, including the
corresponding formulas.
Hint 2: If in this part you have a MIMO system, then you need to use the block diagram and derivations
from Lecture 8a, pages 15-16. Please note that feedback is from the state variables ( )x t not from the
controlled output ( )z t r= . If you have a SISO system then you can use a non-zero tracking result formula
from Lecture 5b (based on Section 3.1.3 from the textbook) , page 9, and the corresponding block diagram.
Design of Optimal Linear-Quadratic (LQ) Controllers
b) 5 pts. Non-Zero Set Point Optimal LQ Controller. Consider the system from part a) or take any
other linear controllable system that you are interested in. Choose the weighed (penalty) matrices such that
the positive definite stabilizing solution of the algebraic Riccati equation exists (see Lecture 6a; textbook
Section 3.5.4). Find the optimal feedback gain and optimal performance values when all initial conditions
are equal to 1. Choose the desired non-zero value for the system output. Follow presentation from Lecture
8a, pages 15-17, and design the Simulink block diagram for the corresponding controller. Plot the obtained
optimal state trajectories that show that the system response settles down at the desired value.
c) 5 pts. Optimal LQ Controller with the Prescribed Degree of Stability. Consider the system from
Part b) or take any other linear controllable system. Based on information obtained in Part b) regarding the
optimal closed-loop optimal eigenvalues, design the optimal LQ controller with the prescribed degree of
stability (Lecture 6b) that moves all closed-loop eigenvalues by two units to the left (deeper into the stability
region,
α = >2 0 ). Plot the obtained optimal closed-loop system response and compare it to the response
obtained in Part b). Observe the impact of the eigenvalue shift on the transient response and the feedback
gain. Assume that all initial conditions are equal to 1. Find the optimal performance criterion.
d) 5 pts. Non-Set Point Optimal LQ Controller with Constant Disturbance. Assume that the system
in Part b) is subjected to a constant input disturbance. An integrator in the feedback loop will be needed to
reject the impact of the constant disturbance so that the desired non-zero output signal can maintain the
same value as in Part b). Design the corresponding LQ optimal controller by following presentation in
Lectures 7b and 8b. Build the corresponding SIMULINK block diagram and plot the results obtained.
Compare the obtained output signal with the system output from Part b).
Hint: See the general detailed derivations of this technique in Lecture 8c (not presented in class, it has
just been posted), where optimal control has (in addition to a constant feedforward term) two feedback
terms: one from the system state variables and another one from the integral of the system output error
( ) ( )desiredy t y t− .
Comment: This problem could have been solved using the presentation of Lecture 7a (Section 3.7 from
the textbook), in which case optimal control will have only one feedback from the system state variables,
in addition to a constant feedforward term.
Required Format for the Project Report
The report should contain the following parts:
1) Project formulation.
2) Provide a brief introduction of the techniques used, including all relevant math formulas.
3) Plot all signals and figures as required by the project. Plot the figures from the MATLAB window
and not from SIMULINK scopes.
4) Include block diagrams (generated by Simulink or any other software) in the main text.
5) Provide comments on the obtained results.
6) Write conclusions.
7) Provide a list of references.
8) Appendix: MATLAB code, and anything else that you consider relevant to the project.