## Weekly outline

### 20 February - 26 February

**Numerical optimization - analysis**- unconstrained optimization
- first-order necessary conditions (scalar and vector case): gradient, directional derivative
- second-order necessary conditions: positive semidefiniteness of Hessian
- second-order sufficient conditions: positive definiteness of Hessian

- constrained optimization
- equality-type constraints: Lagrange multipliers
- inequality-type constraints: KKT conditions

- classes of optimization problems
- linear programming
- quadratic programming (linearly constrained, quadratically constrained)
- (general) nonlinear programming
- other: second-order cone programming, semidefinite programming, ...

Material for the seminar

- unconstrained optimization
### 27 February - 5 March

**Numerical optimization - algorithms**Unconstrained optimization

- Derivative-free methods (Nelder-Mead)
- Derivative-based methods
- Descent direction methods (Gradient method, Newton and Quasi-Newton method)
- Trust region methods (Newton method)

Constrained optimization

- Active set methods
- ...

Material for the seminar

### 6 March - 12 March

**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.

Material for the seminar

### 13 March - 19 March

**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

Material for the seminar

### 20 March - 26 March

**Dynamic programming, approximate dynamic programming, reinforcement learning**- 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
- ...

Material for the seminar

### 27 March - 2 April

**Continuous-time optimal control, calculus of variations, LQ-optimal control**- introduction to calculus of variations: functional, variation of a functional, finite-interval fixed-ends problem, Euler-Lagrange equation as a first-order necessary condition of optimality.
- continuous-time optimality principle - HJB equation

Material for the seminar

### 3 April - 9 April

**Continuous-time optimal control with free final time and constrained inputs, time-optimal control**- calculus of variations for free final time
- minimum-time LQ-optimal control
- minimum-time optimal control under constrained control - transition from calculus of variations to Pontryagin's principle of maximum; bang-bang control for a double integrator and harmonic oscillator
- proximate time-optimal control (PTOS)

Material for the seminar

### 10 April - 16 April

**Numerical methods for continuous-time optimal control**- Indirect approaches to numerical optimal control - solving BVP

- Shooting and multiple shooting (iterating over the unknown boundary conditions)
- Gradient method (minimization of H by iterating over u)
- Quasi-linearisation

- Direct approaches to numerical optimal control - transcribing the optimal control problem into a nonlinear programming problem:
- Direct transcription
- Direct collocation

- Software for numerical optimal control: Acado, ...

Material for the seminar

- Indirect approaches to numerical optimal control - solving BVP
### 17 April - 23 April

**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)

Material for the seminar

**No homework assignment this week!**

### 24 April - 30 April

**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

Material for the seminar

### 1 May - 7 May

**Classical and modern robust control design methods in frequency domain**- Loopshaping (lead, lag, lead-lag, ...)
- 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)

Material for the seminar

### 8 May - 14 May

**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

Material for the seminar

- SISO systems
### 15 May - 21 May

**Linear matrix inequalities (LMI) for control design, synthesis for Linear parameter-varying (LPV) systems -**Rector's Day - Classes canceled!- Linear matrix inequality
- Semidefinite program

Material for the seminar

### 22 May - 28 May

**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

Material for the seminar