Tartalomjegyzék

Advanced Nonlinear Control Methods: Theory and applications (summer school), University of Pannonia, Veszprém

Summer school - 2018

Required software tools

Basic tools

• Matlab - Simulink
• Control System Toolbox
• Symbolic Math Toolbox

LMI tools (you can use tbxmanager)

• YALMIP with
• COIN-OR Linear Program (CLP) Solver (for linear and quadratic problems)
• GLPK solver (for mixed integer problems)
• SeDuMi solver (for semidefinite problems)
• Other solvers: MPT

urlwrite('http://www.tbxmanager.com/tbxmanager.m', 'tbxmanager.m');
tbxmanager install clpmex glpkmex mpt mpt2 mptdoc sedumi yalmip
savepath
!echo "tbxmanager restorepath" > startup.m

If an error occurs with YALMIP, simply download it from the GitHub, unzip it, and add it to the path.

LPV tools

• LPVTools
• Robust Control Toolbox (required by LPVTools)

Material

1st Matlab practice (basic tools)

1. Install YALMIP and SeDuMi
2. Basic symbolic operations - Symbolic Math Toolbox
3. System linearization around equilibrium point - Symbolic Math Toolbox
4. Pole placement (with place) and LQR (with lqr) for the linearized model - Control System Toolbox
5. Static feedback design for the linearized model with LMIs - YALMIP and SeDuMi
6. Static feedback design for an LPV system with LMIs - YALMIP and SeDuMi

2nd Matlab practice (nonlinear control methods)

1. Lie derivative and Lie bracket - Symbolic Math Toolbox
2. Nonlinear state transformation - Symbolic Math Toolbox
3. Nonlinear dynamical systems' simulation with Simulink
4. Solving general transport PDE with Mathematica (​only presentation)
5. Feedback linearization, zero dynamics
6. DOA estimation with quadratic Lyapunov functions (Van der Pol system)

Homework exercises and Matlab sources: here.

Demonstrations and code samples

If you are not an expert in Matlab, consider the following short (Hungarian) tutorial Matlab segédlet.

For advanced users of Symbolic Math Toolbox: deep_symbolic_tricks.

Matlab demonstrations and code samples:

1. 1st Matlab practice: inverted pendulum linearization (Symbolic Math Toolbox) and pole placement
2. 1st Matlab practice: inverted pendulum linearization (with uncertain frictional coefficient) - feedback design for LPV with LMIs
3. 2nd Matlab practice: Geometrical meaning of the Lie derivative and the Lie bracket
4. 2nd Matlab practice: Coordinate transformation, feedback linearization and zero dynamics.
5. 2nd Matlab practice: Estimate DOA with a Lyapunov function
6. Embedding of a rational model into a polynomial model

Mathematica demonstrations:

Compute the integral of the PDE appearing in the Frobenius theorem

Some Matlab helper functions

1. vekanal_subsmesh_demo
2. vekanal_quiver_sym_demo

Useful Matlab functions and toolboxes

Built-in Matlab functions:

• ode45

Important function of the Control System Toolbox:

• ss - State space model
• tf - Transfer function model
• place - Pole placement controller design
• lqr - Linear quadratic regulator design
• lqi - Linear quadratic integral regulator design
• pzmap - Poles and zeros
• impulse - Impulse response function
• step - Step response function

Important function of the Robust Control Toolbox:

• ureal
• wcgain - Worst case gain

Important functions of Matlab's Symbolic Math Toolbox (SMT):

• syms a x y z real; syms f(x) g(x,y,z) - Declare (create) multiple scalar symbolic variables or functions
• A = sym('a_%d%d',[3 4]) - Create a single symbolic matrix, vector or scalar variable
• f_sym = x^2+y*z; symvars(f_sym) - List symbolic variables appearing in the object passed as an argument
• subs(f_sym, [x z], [x+y x^2]) - Substitute variables or expression by other variables or expression in a symbolic expression
• f_fh = matlabFunction(f_sym, 'vars', {x [y z]}) - Generate a function handle from a symbolic expression
• in(.., type) - Declare that an variables belongs to a collection (i.e. set) of variables, eg. 'real', 'positive', 'integer', 'rational'
• assume(x > 2 & in(y,'rational')) - See: assumeAlso. Clear every other assumptions on the same variables
• assumAlso(in(x,'integer') & in(y,'positive')) - Introduce more assumptions about the symbolic objects
• assumptions - List existing assumptions.
• partfrac - Partial fractional decomposition of a rational function (useful when producing inverse Laplace transformation of a transfer function). FactorMode: real/complex/full/rational. See also its numerical pair: residue.

Other useful functions of Matlab's Symbolic Math Toolbox (SMT):

• combine(sin(x)+cos(x),'sincos') - May be useful when manipulating trigonometric functions
• charpoly - characteristic polynomial of a matrix

Control systems demonstrations

Inverted pendulum model (inverz inga modell)

1. Model description and derivation using calculus of variations: coming spoon.
2. Model linearization around stable and unstable equilibrium point, simulation and analysis.
3. Framework for the first Matlab practice
4. Model description and task sheet for the first Matlab practice: ccs_gyak08_matlabgyak1.pdf
5. Framework for the second Matlab practice
6. Model description and task sheet for the first Matlab practice: ccs_gyak08_matlabgyak2.pdf
7. Inverted pendulum control and integral reference tracking.
8. Inverted pendulum control and integral reference tracking (augmented, corrected).
9. Inverted pendulum nonlinear control - 2018.07.30. (július 30, hétfő), 11:02
10. Advanced Nonlinear Control Methods (PhD course): Inverted pendulum linearization (Symbolic Math Toolbox) and pole placement
11. Advanced Nonlinear Control Methods (PhD course): Inverted pendulum linearization (with uncertain frictional coefficient) - feedback design for LPV with LMIs

Crane model (rakodó daru modellje)

1. Model description and derivation using calculus of variations (ccs_model_crane.pdf)
2. State space model derivation using symbolic computations
3. Simulink model and simulation (without control): sim_nonlinear_model_demo