9. Some extensions: LQG, LTR, H2
H2-optimal control
B(E)3M35ORR – Optimal and Robust Control
0. Introduction
What is optimal and robust control?
Course outline
Rules of the course
Software for the course
Literature for the course
Similar courses
Projects
1. Optimization – theory and algorithms
Learning goals
Optimization problems
Problem reformulations
Conditions of optimality
Optimization modelling languages
Computing the derivatives
Numerical algorithms
Numerical solvers
References
Homework
2. Discrete-time optimal control – direct approach
Learning goals
General finite-horizon nonlinear optimal control as an NLP
Finite-horizon LQR as a QP
Model predictive control (MPC)
Numerical solvers for MPC
References
Homework
3. Discrete-time optimal control – indirect approach
Learning goals
General finite-horizon nonlinear optimal control as a TP-BVP
Discrete-time LQR on a finite horizon
Discrete-time LQR on an infinite horizon using the DARE
Reference tracking using LQR
Discrete-time algebraic Riccati equation (DARE)
Software for discrete-time LQR and DARE
References
Homework
4. Discrete-time optimal control – dynamic programming (DP)
Learning goals
DP and discrete-time optimal control
Tabular outcomes of DP
LQR via DP
LQ tracking via DP
Differential dynamic programming (DDP)
References
Homework
5. More on MPC
Recursive feasibility
Stability of MPC
Economic MPC
Explicit MPC
References
Homework
6. Continuous-time optimal control - indirect approach via calculus of variations
Learning goals
Overview of continuous-time optimal control
Calculus of variations
General finite-horizon nonlinear optimal control as a TP-BVP
LQR on a finite time horizon
LQR on an infinite time horizon using the CARE
On notation for Hamiltonians and variations
Trajectory stabilization and neigboring extremals
Software for continuous-time LQR and CARE
References
Homework
7. Continuous-time optimal control - indirect approach via Pontryagin’s maximum principle
Learning goals
Pontryagin’s maximum (or minimum) principle
Optimal control with a free final time
Time-optimal control
References
Homework
8. Numerical methods for continuous-time optimal control - both indirect and direct approaches
Learning goals
Numerical methods for indirect approach
Numerical methods for direct approach
Software
References
Homework
8.X. Continuous-time optimal control – dynamic programming
Dynamic programming for continuous-time optimal control
Using HJB equation to solve the continuous-time LQR problem
Differential dynamic programming (DDP)
References
Homework
9. Some extensions: LQG, LTR, H2
Learning goals
Guaranteed GM and PM for LQR
LQR for stochastic systems
LQG control
Loop transfer recovery (LTR)
H2-optimal control
Software
References
Homework
10. Uncertainty modelling and robustness analysis
Learning goals
Uncertainty (in) modelling
Robustness analysis for unstructured uncertainty
Robustness analysis for structured uncertainty
Software
References
Homework
11. Robust control
Learning goals
Mixed sensitivity design
H-infinity optimal control
Software
References
Homework
12. Robust control II.
Learning goals
mu synthesis
H-infinity loop-shaping
Software
References
Homework
13. Limitations of achievable performance
Learning goals
Limitations for SISO systems
Limitations for MIMO systems
References
Homework
14. Model and controller order reduction
Learning goals
Model order reduction
Controller order reduction
Benchmarks
References
Homework
9. Some extensions: LQG, LTR, H2
H2-optimal control
H2-optimal control
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Loop transfer recovery (LTR)
Software