16 February - 22 February
Motivation for optimal and robust control. Introduction to optimization; optimization without constraints
- Discussion of the needs for optimal and robust control.
- General optimization problem; classification into linear, quadratic, ..., general nonlinear programming.
- General unconstrained optimization problem. Assumptions about the optimized functional: real independent variable (no integer programming in this course), smoothness of the optimized functional, convexity.
- Necessary conditions of optimality of the first-order and second-order. Sufficient conditions of optimality of the second order. Taylor expansion. Big-O and little o concepts. Gradient. Hessian. Weierstrass' theorem on existence of a global optimum.
23 February - 1 March
- Analysis of optimality for equality type constraints. Lagrange multiplier. Regularity conditions.
- Analysis of optimality for inequality typ constraints. Karush-Kuhn-Tucker conditions (KKT).
- Numerical solution to a nonlinear optimization problem
- Software for numeric optimization.
2 March - 8 March
Optimal control for a general discrete-time LTI systems; discrete-time LQ optimal control
- formulation of the general problem of optimal control, practical criteria of optimality (integral: IAE, ITAE, ISE, ...),
- necessary conditions of optimality for a general (nonlinear, time-varying) discrete-time dynamical system.
- Optimal control for a linear system minimizing a quadratic cost function (LQ optimal control): finite time horizont, fixed and free final state leading to Lyapunov and Riccati difference equations, respectively.
9 March - 15 March
Discrete-time LQ optimal control - extension from a finite to an infinite control horizon
- Algebraic Riccati equation (ARE). Condititions of existence and uniqueness of stabilizing controller obtained from the solutions to ARE
- Properties of Hamilton matrix - symmetry of eigenvalues with respect to the stability boundary (the unit circle)
- Frekvenční/faktorizační přístup k návrhu LQ-optimálních regulátorů, symmetric root locus.
16 March - 22 March
Variational calculus approach to optimal control for a continuous-time system
- Preview of infinite-dimensional optimization: norms of functions, strong minimum, weak minimum. Functional. Variation of a functional,
- A few problems solved by calculus of variations: minimum distance problem, Dido's problem, brachistochrone problem,
- Basic calculus of variations problem on a finite and fixed interval with fixed ends. Euler-Lagrange equation,
- Basic calculus of variations problem on a finite and fixed interval with one free end ,
- Using Euler-Lagrange equations to find the necessary conditions of optimality for a general optimal control problem. Specialization of the result for LTI systems and a quadratic criterion - LQ optimal control on a finite horizon (both fixed and free final state cases).
23 March - 29 March
Optimal control with free final time and constraints on amplitude of control signals
- Pontryagyn's principle of maximum (minimum).
- Time-optimal control: bang-bang control.
30 March - 5 April
- Bellman's principle of optimality
- Using dynamic programming to derive LQ optimal regulator for a discrete-time LTI system
- Using dynamic programming to derive LQ optimal regulator for a discrete-time LTI system - Hamilton-Jacobi-Bellman (HJB) equation and its solution
6 April - 12 April
LQG, LTR and H2 optimal control
- Stochastic LQ optimal state feedback
- Kalman filter - an optimal estimator of a state
- LQG optimal regulator: designing LQ-optimal state feedback and an optimal estimator (Kalman filter) separately and then putting them toghether/li>
- Unguaranteed robustness of LQG regulators, LTR control (Loop transfer recovery)
- H2 optimal control (minimizing the H2 system norm)
13 April - 19 April
Uncertainty and robustness, analysisis of robust stability and performance
- Uncertainy in models of dynamic systems:
- uncertainty in physical parameters,
- dynamic uncertainties (uncertainties described in frequency domain)
- Analysis of robust stability and robust performance
- H∞ norm of a system
- Structured singular value (SSV, mu)
- Small gain theorem
- Uncertainy in models of dynamic systems:
20 April - 26 April
Design of a robust controller by H∞ optimization
- Controller design by minimizing mixed sensitivity
- Controller design by minimizing the H∞ norm of a generalized system
- Mu synthesis using the algorithm of DK iterations
27 April - 3 May
Analysis of achievable performance
- SISO systems:
- waterbed effect (Bode's integral conditions)
- interpolation conditions for internal stability
- MIMO systems:
- conditioning of systems
- directions of zeros and poles
- relative gain array (RGA) as an indicator of ill-conditioning
- SISO systems:
4 May - 10 May
Design of a robust controller using H∞ Loopshaping
- Normalized coprime factorization of an LTI system
- Controller with two degrees of freedom
- Implementation with antiwindup provisions
11 May - 17 May
Rector's day - the lecture and the exercise are cancelled
18 May - 24 May
Model predictive control (MPC)
Lecture given by Dr. Jaroslav Pekař (HPL)
- Introduction and motivation
- Formulation of a linear predictive controller
- Analysis of properties (receding horizont, stability, robustness, ...)
- Formulation of a nonlinear predictive controller
25 May - 31 May1. zkouška
1 June - 7 June2. zkouška
8 June - 14 June3. zkouška
6 July - 12 July
Model order reduction, controller order reduction
- Basic methods of model order reduction: truncation, residualization
- Reduction of a balanced realization
- Hankel optimal reduction
- Frequency weighted reduction
- Controller order reduction
13 July - 19 July
Semidefinite programming, linear matrix inequalities and applications in control design
- What is a semidefinite program, what is a linear matrix inequality?
- Convexity of the set of solutions to an LMI
- Examples of LMIs outside and inside the field of control theory
- Quadratic stability and stabilization
- Solving the H∞ optimal control problem via LMI, comparing with the solution based on Riccati equations
- Design of H∞ optimal controller for systems with varying parameters (LPV)