Optimal and Robust Control
BE3M35ORR + B3M35ORR + BE3M35ORC
This course is part of an already archived semester and is therefore readonly.
Weekly outline


(Mathematical) optimization – modeling and analysis
Classes of optimization problems
 Linear programming
 Quadratic programming (with linear constraints)
 Conic programming (linear, quadratic with quadratic constraints, semidefinite, ...)
 Nonlinear programming
(Re)formulations of optimization problems
 Tips and tricks
 Absolute values, max elements, ...
 Using software
 Matlab: CVX, Optimization Toolbox, YALMIP
 Julia: JuMP, Convex
 Python: cvxpy
Conditions of optimality (derivativebased, necessary, sufficient)
 Unconstrained optimization
 Gradient
 Hessian
 Constrained optimization
 Lagrangian, Lagrange multipliers, Projected Hessian
 KKT conditions

Numerical optimization – algorithms (derivativebased)
Computing the derivatives
 Finite difference (FD) numerical approximations
 Algorithmic (automatic) differentiation (AD)
Unconstrained optimization
 Descent direction methods
 Gradient (steepestdescent) method
 Newton method
 QuasiNewton method
Constrained optimization
 Active set methods (projected gradient method)

Discretetime optimal control – direct approach, model predictive control (MPC)
 Introduction to optimal control: motivation, optimization criteria (or performance indices), optimization "variables" (controller parameters or control signals).
 Discretetime control for a linear system with a quadratic performance index over a finite time horizon formulated as a quadratic program > openloop control.
 Model predictive control (MPC) aka receding horizon control as a realtime optimizationbased feedback control scheme: regulation, tracking, both simultaneous and sequential formats, soft constraints, practical issues.

Discretetime optimal control – indirect approach, LQoptimal control
 conditions of optimality for a general nonlinear discretetime system  twopoint boundary value problem
 discretetime LQoptimal control on a finite time horizon, initial and final states fixed
 discretetime LQoptimal control on a finite time horizon, final state free: discretetime (recurrent) Riccati equation
 discretetime LQoptimal control on an infinite time horizon  LQoptimal constant state feedback: discretetime algebraic Riccati equation (ARE)
 discretetime LQoptimal tracking and other LQ extensions

Dynamic programming and optimal control
 Bellman's optimality principle
 dynamic programming approach to problems with discrete and finite time and discrete and finite state space
 dynamic programming used to derive LQoptimal controller
 ...

Continuoustime optimal control – indirect approach via calculus of variations, LQoptimal control
 Introduction to calculus of variations
 Functional, variation of a functional
 Finiteinterval fixed and free end problems
 EulerLagrange equation as a firstorder necessary condition of optimality
 General continuoustime optimal control problem
 Control Hamiltonian
 State, costate and stationarity equations (aka control Hamiltonian canonical equations) and boundary conditions as the necessary condition of optimality.
 Continuoustime LQoptimal control problem
 State, costate and stationarity equations and boundary conditions as the necessary conditions of optimality
 Free final state case – differential Riccati equation
 Infinite timehorizon continuoustime LQ optimal control
 Algebraic Riccati equation (ARE)
 Introduction to calculus of variations

Continuoustime optimal control with free final time and constrained inputs, timeoptimal control
 Calculus of variations for free final time
 Minimumtime optimal control under constraints – transition from calculus of variations to Pontryagin's principle of maximum
 Timeoptimal (bangbang) control for a double integrator and a harmonic oscillator
 Proximate timeoptimal control (PTOS)

Numerical methods for continuoustime optimal control
 Direct and indirect methods
 Shooting, multipleshooting and collocation methods
 Software for numerical optimal control: Acado, ...

LQGoptimal control, H2optimal control, Loop Transfer Recovery (LTR)
 LQoptimal control for stochastic systems (random initial state, stochastic disturbance)
 Optimal estimation
 LQGoptimal control
 H2optimal control
 Loop Transfer Recovery (LTR)

Models of uncertainty, analysis of robustness
 uncertainties in real physical parameters
 uncertainty formulated in frequency domain
 unstructured frequency domain uncertainty represented by \(\Delta\) term and a weighting filter W
 structured frequencydomain uncertainty
 additive, multiplicative, inverse models of uncertainty
 small gain theorem based robust stability and robust performance analysis

Classical and modern robust control design methods in frequency domain
 Loopshaping (lead, lag, leadlag, ...)
 Quantitative Feedback Theory (QFT)
 \(\mathcal{H}_\infty\)minimizationbased control design
 standard \(\mathcal{H}_\infty\)optimal control
 mixed sensitivity minimization
robust loopshaping (assuming coprime factor uncertainty)
\(\mu\) synthesis (DK iterations)

Analysis of limits of achievable performance
 SISO systems
 Scaling
 Integral constraints
 Interpolation constraints
 Limitations due to delay
 Limitations due to disturance
 Limitations due to saturation of controls
 MIMO systems
 Directionality of MIMO systems
 Illconditioning of MIMO systems
 Relative Gain Array (RGA)
 Limitations due to uncertainty
 SISO systems

Model and controller order reduction
 Basic order reduction techniques: truncation and residualization
 Balanced statespace realization: simultaneous diagonalization of observability and controllability gramians
 Balanced truncation / balanced residualization
 Hankel norm minimization
 Frequencyweighted approximation and stabilityguaranteeing controllerorder reduction

Linear Matrix Inequalities (LMI) for control analysis and design, Semidefinite Programming (SDP), control synthesis for Linear ParameterVarying (LPV) systems