## Weekly outline

• ### 20 February - 26 February

#### (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 (derivative-based, necessary, sufficient)
• Unconstrained optimization
• Hessian
• Constrained optimization
• Lagrangian, Lagrange multipliers, Projected Hessian
• KKT conditions

• At home

• At the “lecture"

• Material for the exercises

• ### 27 February - 5 March

#### Numerical optimization – algorithms (derivative-based)

##### Computing the derivatives
• Finite difference (FD) numerical approximations
• Algorithmic (automatic) differentiation (AD)

##### Unconstrained optimization
• Descent direction methods
• Gradient (steepest-descent) method
• Newton method
• Quasi-Newton method

##### Constrained optimization
• Active set methods (projected gradient method)
• At home

• At the “lecture"

• Material for the exercises

### Discrete-time 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).
• Discrete-time control for a linear system with a quadratic performance index over a finite time horizon formulated as a quadratic program -> open-loop control.
• Model predictive control (MPC) aka receding horizon control as a real-time optimization-based feedback control scheme: regulation, tracking, both simultaneous and sequential formats, soft constraints, practical issues.
• At home

• At the “lecture"

• Material for the exercises

### Discrete-time optimal control – indirect approach, LQ-optimal control

• conditions of optimality for a general nonlinear discrete-time system - two-point boundary value problem
• discrete-time LQ-optimal control on a finite time horizon, initial and final states fixed
• discrete-time LQ-optimal control on a finite time horizon, final state free: discrete-time (recurrent) Riccati equation
• discrete-time LQ-optimal control on an infinite time horizon - LQ-optimal constant state feedback: discrete-time algebraic Riccati equation (ARE)
• discrete-time LQ-optimal tracking and other LQ extensions
• At home

• At the "lecture"

• Material for the seminar

### 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 LQ-optimal controller
• ...
• At home

• At the "physical lecture"

• Material for the seminar

### Continuous-time optimal control – indirect approach via calculus of variations, LQ-optimal control

• Introduction to calculus of variations
• Functional, variation of a functional
• Finite-interval fixed- and free end problems
• Euler-Lagrange equation as a first-order necessary condition of optimality
• General continuous-time optimal control problem
• Control Hamiltonian
• State, costate and stationarity equations (aka control Hamiltonian canonical equations) and boundary conditions as the necessary condition of optimality.
• Continuous-time LQ-optimal control problem
• State, costate and stationarity equations and boundary conditions as the necessary conditions of optimality
• Free final state case – differential Riccati equation
• Infinite time-horizon continuous-time LQ optimal control
• Algebraic Riccati equation (ARE)
• At home

• At the "physical lecture"

• Material for the seminar

### Continuous-time optimal control with free final time and constrained inputs, time-optimal control

• Calculus of variations for free final time
• Minimum-time optimal control under constraints – transition from calculus of variations to Pontryagin's principle of maximum
• Time-optimal (bang-bang) control for a double integrator and a harmonic oscillator
• Proximate time-optimal control (PTOS)
• At home

• Material for the seminar

### Numerical methods for continuous-time optimal control

• Direct and indirect methods
• Shooting, multiple-shooting and collocation methods
• Software for numerical optimal control: Acado, ...
• At home

• Material for the seminar

### LQG-optimal control, H2-optimal control, Loop Transfer Recovery (LTR)

• LQ-optimal control for stochastic systems (random initial state, stochastic disturbance)
• Optimal estimation
• LQG-optimal control
• H2-optimal control
• Loop Transfer Recovery (LTR)
• At home

• Material for the seminar

• No homework assignment this week!

### 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 frequency-domain uncertainty
• additive, multiplicative, inverse models of uncertainty
• small gain theorem based robust stability and robust performance analysis
• At home

• Material for the seminar

• No homework assignment this week!

### Classical and modern robust control design methods in frequency domain

• Quantitative Feedback Theory (QFT)
• $$\mathcal{H}_\infty$$-minimization-based control design
• standard $$\mathcal{H}_\infty$$-optimal control
• mixed sensitivity minimization
• robust loopshaping (assuming coprime factor uncertainty)
• $$\mu$$ synthesis (DK iterations)
• At home

• Material for the seminar

### 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
• Ill-conditioning of MIMO systems
• Relative Gain Array (RGA)
• Limitations due to uncertainty
• At home

• Material for the seminar

### Model and controller order reduction

• Basic order reduction techniques: truncation and residualization
• Balanced state-space realization: simultaneous diagonalization of observability and controllability gramians
• Balanced truncation / balanced residualization
• Hankel norm minimization
• Frequency-weighted approximation and stability-guaranteeing controller-order reduction
• At home

• Material for the seminar

### Linear Matrix Inequalities (LMI) for control analysis and design, Semidefinite Programming (SDP), control synthesis for Linear Parameter-Varying (LPV) systems

• Material for the seminar