Optimal and Robust Control
BE3M35ORR + B3M35ORR + BE3M35ORC
Tento kurz je součástí již archivovaného semestru, a proto je dostupný pouze pro čtení.
Bohužel, tato činnost je momentálně skrytá
Osnova sekce
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(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
- Gradient
- Hessian
- Constrained optimization
- Lagrangian, Lagrange multipliers, Projected Hessian
- KKT conditions
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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)
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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.
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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
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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
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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)
- Introduction to calculus of variations
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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)
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Numerical methods for continuous-time optimal control
- Direct and indirect methods
- Shooting, multiple-shooting and collocation methods
- Software for numerical optimal control: Acado, ...
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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)
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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
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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)
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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
- SISO systems
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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
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Linear Matrix Inequalities (LMI) for control analysis and design, Semidefinite Programming (SDP), control synthesis for Linear Parameter-Varying (LPV) systems