Official website of the course
Important built-in linear algebra functions:
- Q = orth(A) - Gives an orthonormal basis of the image space of a matrix
- Z = null(A) - Returns an orthonormal basis for the null space of matrix A. W = null(V') returns an orthonormal basis of the orthogonal complement of subspace V spanned by the column vectors collected in matrix V
- [U,S,V] = svd(A) - Singluar value decomposition of matrix A (see also Fundamental theorem of linear algebra)
- k = rank(A,
) - Rank of matrix A- [R,p] = rref(A) - Reduced row echelon form (Gauss-Jordan elimination). p gives the indices of the first k nr of linearly independent rows of matrix A, where k = rank(A)
- [r,p,k] = residue(b,a); [b,a] = residue(r,p,k) - Partial fractional decomposition (numerical)
- w = conv(u,v) - Convolution and polynomial multiplication
- [q,r] = deconv(u,v) - Deconvolution and polynomial division
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
- simplify - Simplify symbolic expression
- simplifyFraction - Simplify fraction of polynomials expression to its irreducible form
- 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
- charpoly - Compute the characteristic polynomial of a matrix
- 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.
- laplace
- ilaplace
Important function of the Control System Toolbox:
- [b,a] = ss2tf(A,B,C,D) - State-space model to transfer function model
- [A,B,C,D] = tf2ss(b,a) - Transfer function model to state-space model
- ss - State space model
- tf - Transfer function model
- impulse(sys) - Impulse response function
- step(sys) - Step response function
- lsim(sys,u,t,x0) - Simulate an LTI system (sys) with arbitrary input u(t) from initial condition x0
- On = obsv(A,B) - Compute the observability matrix of LTI system (A,B,C,D)
- Cn = ctrb(A,B) - Compute the controllability matrix of LTI system (A,B,C,D)
Important built-in linear algebra functions:
- r = rank(A) - Compute the rank of matrix A
- Ker = null(A) - Compute an orthonormal basis for the kernel space of matrix A
- Im = orth(A) - Compute an orthonormal basis for the image space of matrix A using Gram-Schmidt orthogonalization
- A_GJ = orth(A) - Compute the reduced row echelon form of matrix A (Gauss-Jordan elimination)
Matlab examples
syms s
H = (5*s^2 + 3*s + 1)/(s^3 + 6*s^2 + 11*s + 6);
partfrac(H)
ans = 3/(2*(s + 1)) - 15/(s + 2) + 37/(2*(s + 3))$H(s) = \frac{3}{2(s+1)}-\frac{15}{s+2}+\frac{37}{2(s+3)}$
syms t a b c e f
h = cos(a*t) + sin(a*t) + exp(-c*t)*cos(b*t) + exp(-f*t)*sin(e*t)
laplace(h,t)
ans = a/(a^2 + t^2) + (c + t)/((c + t)^2 + b^2) + e/((f + t)^2 + e^2)$H(s) = \frac{a}{a^2+t^2}+\frac{t}{a^2+t^2}+\frac{c+t}{{(c+t)}^2+b^2}+\frac{e}{{(f+t)}^2+e^2}$
syms s
ilaplace(1/(s^2+s+1))Matlab examples
b = [1 0 1]; a = [1 0 2 3 2];
H = tf(b,a)H =
s^2 + 1
---------------------
s^4 + 2 s^2 + 3 s + 2
Continuous-time transfer function.
sys = ss(H)sys =
A =
x1 x2 x3 x4
x1 0 -1 -1.5 -1
x2 2 0 0 0
x3 0 1 0 0
x4 0 0 1 0
B =
u1
x1 1
x2 0
x3 0
x4 0
C =
x1 x2 x3 x4
y1 0 0.5 0 0.5
D =
u1
y1 0
Continuous-time state-space model.
[A,B,C,D] = ssdata(sys)A =
0 -1.0000 -1.5000 -1.0000
2.0000 0 0 0
0 1.0000 0 0
0 0 1.0000 0
B =
1
0
0
0
C =
0 0.5000 0 0.5000
D =
0
A = randn(6); B = randn(6,2); C = randn(3,6); D = zeros(3,2);
sys = ss(A,B,C,D)sys =
A =
x1 x2 x3 x4 x5 x6
x1 -0.1241 0.7172 0.2939 -2.944 -0.1022 -0.1649
x2 1.49 1.63 -0.7873 1.438 -0.2414 0.6277
x3 1.409 0.4889 0.8884 0.3252 0.3192 1.093
x4 1.417 1.035 -1.147 -0.7549 0.3129 1.109
x5 0.6715 0.7269 -1.069 1.37 -0.8649 -0.8637
x6 -1.207 -0.3034 -0.8095 -1.712 -0.03005 0.07736
B =
u1 u2
x1 -1.214 -0.2256
x2 -1.114 1.117
x3 -0.006849 -1.089
x4 1.533 0.03256
x5 -0.7697 0.5525
x6 0.3714 1.101
C =
x1 x2 x3 x4 x5 x6
y1 1.544 -0.7423 -0.6156 0.8886 -1.422 -0.1961
y2 0.08593 -1.062 0.7481 -0.7648 0.4882 1.419
y3 -1.492 2.35 -0.1924 -1.402 -0.1774 0.2916
D =
u1 u2
y1 0 0
y2 0 0
y3 0 0
Continuous-time state-space model.
H = tf(sys)H =
From input 1 to output...
1.34 s^5 - 11.81 s^4 + 51.54 s^3 - 67.91 s^2 + 15.52 s + 8.946
1: ---------------------------------------------------------------------
s^6 - 0.852 s^5 + 2.835 s^4 - 11.81 s^3 + 8.768 s^2 + 5.545 s - 5.529
0.05174 s^5 + 1.64 s^4 + 26.25 s^3 - 30.64 s^2 - 15.04 s + 10.24
2: ---------------------------------------------------------------------
s^6 - 0.852 s^5 + 2.835 s^4 - 11.81 s^3 + 8.768 s^2 + 5.545 s - 5.529
-2.709 s^5 + 12.98 s^4 - 52.9 s^3 + 73.51 s^2 - 0.02814 s - 32.15
3: ---------------------------------------------------------------------
s^6 - 0.852 s^5 + 2.835 s^4 - 11.81 s^3 + 8.768 s^2 + 5.545 s - 5.529
From input 2 to output...
-1.48 s^5 + 1.224 s^4 - 31.77 s^3 + 47.39 s^2 - 4.437 s - 12.79
1: ---------------------------------------------------------------------
s^6 - 0.852 s^5 + 2.835 s^4 - 11.81 s^3 + 8.768 s^2 + 5.545 s - 5.529
-0.2133 s^5 - 3.697 s^4 - 14.09 s^3 + 15.6 s^2 + 4.706 s - 7.059
2: ---------------------------------------------------------------------
s^6 - 0.852 s^5 + 2.835 s^4 - 11.81 s^3 + 8.768 s^2 + 5.545 s - 5.529
3.35 s^5 - 0.9953 s^4 + 37.94 s^3 - 58.62 s^2 - 7.658 s + 25.87
3: ---------------------------------------------------------------------
s^6 - 0.852 s^5 + 2.835 s^4 - 11.81 s^3 + 8.768 s^2 + 5.545 s - 5.529
Continuous-time transfer function.
[b,a] = tfdata(H)b =
3×2 cell array
{1×7 double} {1×7 double}
{1×7 double} {1×7 double}
{1×7 double} {1×7 double}
a =
3×2 cell array
{1×7 double} {1×7 double}
{1×7 double} {1×7 double}
{1×7 double} {1×7 double}
Matlab examples
A = [1 2 -2 ; 0 1 0 ; 1 0 1]; B = [0 0 2]'; C = [1 0 0]; D = 0;
Cn = ctrb(A,B), rank(Cn)Cn =
0 -4 -8
0 0 0
2 2 -2
ans = 2
Ctrb_Subsp = orth(Cn)Ctrb_Subsp =
0.9933 -0.1153
0 0
0.1153 0.9933
Ctrb_Subsp = orth(sym(Cn))Ctrb_Subsp = [ 0, -1] [ 0, 0] [ 1, 0]
On = obsv(A,C), rank(On)On =
1 0 0
1 2 -2
-1 4 -4
ans = 2
Unobsv_Subsp = null(On)Unobsv_Subsp =
0
0.7071
0.7071
Unobsv_Subsp = null(sym(On))Unobsv_Subsp = 0 1 1
Inverted pendulum model (inverz inga modell)
- Model description and task sheet for the first Matlab practice: sem_08_2019_matlab1.pdf
- Framework for the first Matlab practice
- Model description and task sheet for the first Matlab practice: sem_10_2019_matlab2.pdf
- Framework for the second Matlab practice
- Model linearization around stable and unstable equilibrium point, simulation and analysis.
- Inverted pendulum control and integral reference tracking.
- Inverted pendulum control and integral reference tracking (augmented, corrected).
- Inverted pendulum nonlinear control - 2018.07.30. (július 30, hétfő), 11:02
- Advanced Nonlinear Control Methods (PhD course): Inverted pendulum linearization (Symbolic Math Toolbox) and pole placement
- Advanced Nonlinear Control Methods (PhD course): Inverted pendulum linearization (with uncertain frictional coefficient) - feedback design for LPV with LMIs
Crane model (rakodó daru modellje)
- Model description and derivation using calculus of variations (ccs_model_crane.pdf)
- State space model derivation using symbolic computations
- Simulink model and simulation (without control): sim_nonlinear_model_demo
| Andrew_Lewis_A_Mathematical_Approach_to_Classical_Control.pdf | 2731 days | 9.32 MB | ||
| Scherer.Weiland_2000_Linear_Matrix_Inequalities_in_Control.pdf | 2731 days | 1.25 MB | ||
| The_Fundamental_Theorem_of_Linear_Algebra.pdf | 2731 days | 330.57 KB |
| 2015_ccs_potzh_elm.pdf | 2731 days | 101.32 KB | ||
| 2015_ccs_potzh_gyak_MO.pdf | 2731 days | 117.14 KB | ||
| 2016_ccs_potzh_elm.pdf | 2731 days | 96.75 KB | ||
| 2016_ccs_potzh_elm_en.pdf | 2358 days | 100.7 KB | ||
| 2016_ccs_potzh_gyak.pdf | 2731 days | 82.16 KB | ||
| 2016_ccs_potzh_gyak_en.pdf | 2358 days | 84.95 KB | ||
| 2017a_ccs_potzh.pdf | 2731 days | 139.12 KB | ||
| 2017b_ccs_potpotzh .pdf | 2731 days | 117.83 KB | ||
| 2017b_ccs_potzh.pdf | 2731 days | 85.86 KB | ||
| 2018_ccs_potzh_elm.pdf | 2713 days | 91.27 KB | ||
| 2018_ccs_potzh_gyak.pdf | 2713 days | 109.1 KB |
Basic linear algebra, Labplace transformation, solution of ODEs with Laplace transformation.
Sources:
Solved Matlab exercises:
Important built-in linear algebra functions:
Convex optimization, primal-dual problem, introductio to LMIs.
Recommended literature: Scherer.Weiland_2000_Linear_Matrix_Inequalities_in_Control.pdf
Handwritten lecture notes: dark, original
System inversion of minimum phase SISO systems
Observability, controllability, model transformation into/from different model representations. Controllability staircase form.
Dissipativity, stability, estimating induced $\mathcal L_2$ operator gain for LTI and affine LPV systems.
Recommended literature: Scherer.Weiland_2000_Linear_Matrix_Inequalities_in_Control.pdf
Handwritten lecture notes: dark, original
Tasks:
Package for Matlab practice: lab_08_2018_TP_2018_11_06.zip
Things to do:
Published Matlab scripts:
Inverted pendulum model (inverz inga modell)
- Model description and derivation using calculus of variations: coming spoon.
- Model linearization around stable and unstable equilibrium point, simulation and analysis.
- Framework for the first Matlab practice
- Model description and task sheet for the first Matlab practice: ccs_gyak08_matlabgyak1.pdf
- Framework for the second Matlab practice
- Model description and task sheet for the first Matlab practice: ccs_gyak08_matlabgyak2.pdf
- Inverted pendulum control and integral reference tracking.
- Inverted pendulum control and integral reference tracking (augmented, corrected).
- Inverted pendulum nonlinear control - 2018.07.30. (július 30, hétfő), 11:02
- Advanced Nonlinear Control Methods (PhD course): Inverted pendulum linearization (Symbolic Math Toolbox) and pole placement
- Advanced Nonlinear Control Methods (PhD course): Inverted pendulum linearization (with uncertain frictional coefficient) - feedback design for LPV with LMIs
Crane model (rakodó daru modellje)
- Model description and derivation using calculus of variations (ccs_model_crane.pdf)
- State space model derivation using symbolic computations
- Simulink model and simulation (without control): sim_nonlinear_model_demo
Eredmények
| Midterm sample test sheets for the 2nd midterm | ||||
| 2015_ccs_2zh_elm.pdf | 2731 days | 104.75 KB | ||
| 2015_ccs_2zh_gyak_MO.pdf | 2731 days | 114.2 KB | ||
| 2015_ccs_gyakorlo_zh.pdf | 2731 days | 104.09 KB | ||
| 2015_ccs_potzh_gyak_MO.pdf | 2731 days | 117.14 KB | ||
| 2016_ccs_2zh_elm.pdf | 2731 days | 83.64 KB | ||
| 2016_ccs_2zh_gyak.pdf | 2731 days | 103.98 KB | ||
| 2017_ccs_2zh_elm.pdf | 2731 days | 104.28 KB | ||
| 2017_ccs_2zh_gyak.pdf | 2731 days | 131.4 KB | ||
| Lecture notes | ||||
| ccs_gyak01.pdf | 2731 days | 311.19 KB | ||
| ccs_gyak02.pdf | 2731 days | 407.6 KB | ||
| ccs_gyak02_eng.pdf | 2731 days | 261.36 KB | ||
| ccs_gyak03.pdf | 2731 days | 441.73 KB | ||
| ccs_gyak04.pdf | 2731 days | 539.06 KB | ||
| ccs_gyak05.pdf | 2731 days | 459.68 KB | ||
| ccs_gyak06.pdf | 2731 days | 399.69 KB | ||
| ccs_gyak07.pdf | 2731 days | 260.77 KB | ||
| ccs_gyak08_matlabgyak1.pdf | 2731 days | 184.64 KB | ||
| ccs_gyak09.pdf | 2731 days | 150.44 KB | ||
| ccs_gyak10_matlabgyak2.pdf | 2731 days | 390.06 KB | ||
| ccs_gyak11.pdf | 2731 days | 181.51 KB | ||
| ccs_gyakorlo.pdf | 2731 days | 114.22 KB | ||
| Homework exercises | ||||
| ccs_hf1.pdf | 2731 days | 351.62 KB | ||
| ccs_hf1_mo.pdf | 2731 days | 371 KB | ||
| ccs_hf2.pdf | 2731 days | 239.88 KB | ||
| ccs_hf2_mo.pdf | 2731 days | 289.16 KB | ||
| ccs_hf3 másolata.pdf | 2731 days | 224.15 KB | ||
| ccs_hf3.pdf | 2731 days | 224.15 KB | ||
| ccs_hf3_mo.pdf | 2731 days | 231.13 KB | ||
| ccs_hf4.pdf | 2731 days | 94.95 KB | ||
| ccs_lec01.pdf | 2731 days | 445.89 KB | ||
| ccs_lec02.pdf | 2731 days | 263.84 KB | ||
| ccs_lec02_v2.pdf | 2731 days | 261.76 KB | ||
| Model descriptions | ||||
| ccs_model_crane.pdf | 2731 days | 217.4 KB | ||
| ccs_model_double_crane.pdf | 2731 days | 318.02 KB | ||
| ccs_model_gantry_crane_Ospina_Henao.pdf | 2731 days | 686.56 KB | ||
| ccs_segedlet_Matrixok.pdf | 2731 days | 141.88 KB | ||
| Extra material and tutorial | ||||
| segedlet_Laplace_table.pdf | 2731 days | 98.74 KB | ||
| segedlet_szorgalmi_lqservo.pdf | 2731 days | 30.01 KB | ||
| midterm seating | ||||
| zh1_beosztas.pdf | 2731 days | 44.72 KB | ||
| zh2_beosztas.pdf | 2731 days | 44.71 KB |