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| | | | | 2 March - 8 March
Static optimization:- unconstrained optimization, stationary points, local extrema, saddle points, singular points
- Lagrange multipliers
- Hamiltonians
Discrete-time LQ control:- general discrete-time nonlinear optimal controller
- discrete-time linear quadratic regulator:
- state and costate equations, adjoint system, Lyapunov equation
- fixed final state - open loop control
- free final state - closed-loop control
Steady-state discrete-time LQ optimal control:- algebraic Riccati equation and relation to the limiting solution of difference Riccati equation
- conditions on existence of a unique stabilizing LQ controller.
- factorization approach
- Chang-Letov design procedure (symmetric root locus)
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| | | | 9 March - 15 March
Continuous LQ control:- Variations, Leibniz's rule
- General continuous-time optimal control problem: necessary conditions
- Continuous-time LQ-optimal control problem: (matrix) differential Riccati equations
- Steady-state LQ-(sub)optimal control: algebraic Riccati equation
- Eigenvector approach to solution of Riccati equation
- Chang-Letov design procedure (symmetric root locus)
Time-optimal control (bang-bang control):
- General optimal control problems with free final time
- Examples: Zermelo's problem, Brachistochrone problem, Minimum-time injection to the orbit problem
- LQ optimal control enhanced with final time minimization (LQMT)
- Contraint input: Pontryagin principle of minimum
- Bang-bang time-optimal control of linear systems
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| | | | 16 March - 22 March
Analysis of robustness against unstructured dynamic uncertainty:
- uncertainty in modelling, robustness, robust control
- modelling uncertainty:
- uncertainty in real physical parameters (intervals)
- uncertainty in operating point (polytopic systems)
- dynamic uncertainty:
- unstructured (Hinf norm-bounded)
- and structured (mu-bounded)
- types of dynamic uncertainties:
- additive
- multiplicative (input and output)
- inverse (additive, multilicative)
- general: linear fractional transformation (LFT)
- robust stability with multiplicative uncertainty: small gain theorem
- robust stability with a general uncertainty described by LFT
- nominal and robust performance with multiplicative uncertainty
- robust performance cast as robust stability with structured uncertainty
Analysis of robustness against structured dynamic uncertainty (structured singular values):
- Motivation for structured dynamic uncertainty: mixing robust stability and performance, several uncertain blocks
- Definition of structured singular value μ (SSV, mu)
- Computation of μ: lower and upper bounds
- Ex: Casting a robust performance problem as a robust stability
problem against structured uncertainty (two scalar diagonal block and
one diagonal 2x2 "performance" block
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| | | | 23 March - 29 March
Design of robust controllers minimizing mixed sensitivity function, H∞-optimal control:
- Formulation of mixed-sensitivity control design problem
- Achievable shapes of sensitivity and complementary sensitivity functions: limitations imposed by right-half-plane zeros, unstable poles, requirements on integral action, low overshoot, disturbance attenuation.
- Casting mixed sentitivity minimization as a general H∞ minimization problem
- Signal-based H∞ minimization viepoint of a control design
Design of robust controllers minimizing (weighted) structured singular values (mu-synthesis):
- Interpolation with stable transfer functions
- DK iteration
Design of robust controllers by loopshaping (Glover-McFarlane):
- Coprime (stable) factorization of transfer functions
- Coprime factor uncertainty
- Systematic procedure for loopshaping design
- Implementation issues: 1-and-1/2-degree-of-freedom controller, two-degree-of-freedom controller
- Observer structure of "loopshaping" controller
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| | | | 30 March - 5 April
LMI, semidefinite programming:
- Semidefinite programming, linear matrix inequality
- Formats of linear matrix inequalities: canonical and matrix
- Types of LMI problems: feasibility problem, linear optimization, generalized eigenvalue problem
- S-procedure
- Finsler's lemma
- Introduction to (some) software for SDP: LMI Control System Toolbox, Sedumi, Yalmip.
Application of LMI in robust control: quadratic stability, Hinf:
- Quadratic stability, single Lyapunov function for a polytope of LTI systems.
- Bounded real lemma (=LMI bound on Hinf norm of an LTI system)
- Computing mu using LMI optimization
- Stabilizing state feedback for a polytope of LTI systems
(converting bilinear matrix inequality to linear one using change of
variables)
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| | | | 6 April - 12 April
Time-optimal control (bang-bang control):
- General optimal control problems with free final time
- Examples: Zermelo's problem, Brachistochrone problem, Minimum-time injection to the orbit problem
- LQ optimal control enhanced with final time minimization (LQMT)
- Contraint input: Pontryagin principle of minimum
- Bang-bang time-optimal control of linear systems
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| | | | 13 April - 19 April
Application of LMI in robust control: quadratic stability, Hinf:
- Quadratic stability, single Lyapunov function for a polytope of LTI systems.
- Bounded real lemma (=LMI bound on Hinf norm of an LTI system)
- Computing mu using LMI optimization
- Stabilizing state feedback for a polytope of LTI systems (converting bilinear matrix inequality to linear one using change of variables)
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| | | | 20 April - 26 April
Analysis of robustness against structured dynamic uncertainty (structured singular values):
- Motivation for structured dynamic uncertainty: mixing robust stability and performance, several uncertain blocks
- Definition of structured singular value μ (SSV, mu)
- Computation of μ: lower and upper bounds
- Ex: Casting a robust performance problem as a robust stability problem against structured uncertainty (two scalar diagonal block and one diagonal 2x2 "performance" block
As this was a "no show lecture", students are required to study the material in sections 8.6 to 8.11 in [SP2005]. Familiarity with functions lft, ultidyn and mussv is necessary for computation.
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| | | | 27 April - 3 May
Steady-state discrete-time LQ optimal control:- algebraic Riccati equation and relation to the limiting solution of difference Riccati equation
- conditions on existence of a unique stabilizing LQ controller.
- factorization approach
- Chang-Letov design procedure (symmetric root locus)
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| | | | 4 May - 10 May
Discrete-time LQ control:- general discrete-time nonlinear optimal controller
- discrete-time linear quadratic regulator:
- state and costate equations, adjoint system, Lyapunov equation
- fixed final state - open loop control
- free final state - closed-loop control
- Prepared using Chapter 2 from (F. Lewis 1995).
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| | | | 11 May - 17 May
Design of robust controllers minimizing (weighted) structured singular values (mu-synthesis):
- Interpolation with stable transfer functions
- DK iteration
The computations will be performed with Robust Control Toolbox for Matlab, but there is another "free" toolbox from J.F. Magni called Linear Fractional Representation Toolbox (LFRT) and its extension called Skew-mu Toolbox. Give it a try.
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| | | | 18 May - 24 May
Design of robust controllers by loopshaping (Glover-McFarlane):
- Coprime (stable) factorization of transfer functions
- Coprime factor uncertainty
- Systematic procedure for loopshaping design
- Implementation issues: 1-and-1/2-degree-of-freedom controller, two-degree-of-freedom controller
- Observer structure of "loopshaping" controller
The lecture is based on Section 9.4 in [SP2005] (pages 364-382).
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| | | | 25 May - 31 May
Model order reduction:
- Balanced state-space realization
- Controllability gramian, observability gramian
- Hankel operator, singular values of Hankel operator
- Balanced truncation
- Balanced residualization
- Optimal Hankel norm approximation
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| | | | 1 June - 7 June
Uncertainty in modeling, robustness:
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Nothing new since your last login This advanced course will cover modern design methods for optimal and robust control like LQ-optimal control, time-optimal (bang-bang) control, Hinf-optimal control, mixed-sensitivity minimization, DK-iteration, robust loopshaping. Semidefinite programming and linear matrix inequalities (LMI) will be covered as they constitute a an ellegant theoretical and powerful computation tool for solving all those tasks. Emphasis in the course will be put on the actual computational design skills (in Matlab).
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