Tartalomjegyzék

Computer controlled systems (2019)

Official website of the course

Lecture notes of the seminar

Homeworks
hw_1_2019.pdf		1880 days		40.3 KB
hw_2_2019.pdf		1871 days		36.77 KB
hw_3_2019.pdf		1856 days		189.57 KB
Lecture slides
lec_01_2019.pdf		1897 days		4.92 MB
lec_02_2019.pdf		1890 days		461.3 KB
lec_03_2019.pdf		1877 days		905.26 KB
lec_04_2019.pdf		1876 days		401.72 KB
lec_05_2019.pdf		1869 days		1009.69 KB
lec_06_2019.pdf		1862 days		1.3 MB
lec_07_2019.pdf		1855 days		1.63 MB
lec_08_2019.pdf		1841 days		1.09 MB
lec_09_2019.pdf		1834 days		1.03 MB
Seminars (1st midterm)
sem_01_2019.pdf		1883 days		114.78 KB
sem_02_2019.pdf		1896 days		107.51 KB
sem_03_2019.pdf		1882 days		141.71 KB
sem_04_2019.pdf		1882 days		340.58 KB
sem_05_2019.pdf		1871 days		252.81 KB
Seminars (2nd midterm)
sem_06_2019.pdf		1840 days		347.98 KB
sem_06_2019_TP_matlab.zip		1473 days		28.71 KB
sem_08_2019_matlab1.pdf		1833 days		311.2 KB
sem_08_2019_matlab1.zip		1473 days		4.26 KB
sem_08_2019_matlab1_sol.zip		1473 days		4.44 KB
sem_09_2019.pdf		1856 days		209.94 KB
sem_10_2019_matlab2.pdf		1833 days		493.63 KB
sem_10_2019_matlab2.zip		1473 days		3.14 KB
Extra homeworks (not compulsary)
xtra_hw_dynamical_system_simulations.pdf		1833 days		173.68 KB
xtra_hw_linear_algebra.pdf		1833 days		337.11 KB
Extra material and tutorial
xtra_material_Convex_optimization.pdf		2181 days		370.89 KB
xtra_material_DC_motor_model.pdf		2181 days		302.86 KB
xtra_material_Dissipativity_L2gain_LTI_LPV.pdf		2181 days		316.85 KB
xtra_material_Dissipativity_L2gain_LTI_LPV_dark.pdf		2181 days		3.02 MB
xtra_material_Laplace_table.pdf		2181 days		98.74 KB
xtra_material_lqservo.pdf		2181 days		30.01 KB
Model descriptions
xtra_model_crane.pdf		1833 days		217.07 KB
xtra_model_double_crane.pdf		2181 days		318.02 KB
xtra_model_gantry_crane_Ospina_Henao.pdf		2181 days		686.56 KB
midterm seating
zh1_seating.pdf		1861 days		28.51 KB
zh2_seating.pdf		1861 days		26.85 KB

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)

Seminar 1. - Laplace transformation, non-dimensional ODEs

Matlab examples

1. Compute a polynomial division (see lecture notes nr. 3).
2. Compute the partial fractional decomposition of $H(s) = \frac{5s^2+3s+1}{s^3+6s^2+11s+6}$.

   syms s
   H = (5*s^2 + 3*s + 1)/(s^3 + 6*s^2 + 11*s + 6);
   partfrac(H)

   Output:

   ans = 3/(2*(s + 1)) - 15/(s + 2) + 37/(2*(s + 3))

   $H(s) = \frac{3}{2(s+1)}-\frac{15}{s+2}+\frac{37}{2(s+3)}$

3. Compute the Laplace transformation of $h(t) = \cos(at)+\sin(at)+e^{-ct}\cos(bt)+e^{-ft}\sin(et)$.

   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)

   Output:

   ans = a/(a^2 + t^2) + (c + t)/((c + t)^2 + b^2) + e/((f + t)^2 + e^2)

   $H(s) = \frac{a}{a^2+t^2}+\frac{t}{a^2+t^2}+\frac{c+t}{{(c+t)}^2+b^2}+\frac{e}{{(f+t)}^2+e^2}$

4. Compute the inverse Laplave transformation of $H(s) = \frac{1}{s^2 + s + 1}$.

   syms s
   ilaplace(1/(s^2+s+1))

   Output:

   $h(t) = \frac{2\sqrt{3}{e}^{-\frac{t}{2}}\sin(\frac{\sqrt{3}t}{2})}{3}$

Seminar 3. - System representations

Matlab examples

1. Give a possible stat-space realization for LTI system $H(s) = \frac{b(s)}{a(s)}$.

   b = [1 0 1]; a = [1 0 2 3 2];
   H = tf(b,a)

   Output:

   H =
   
            s^2 + 1
     ---------------------
     s^4 + 2 s^2 + 3 s + 2
   
   Continuous-time transfer function.

   sys = ss(H)

   Output:

   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)

   Output:

   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

2. Compute the transfer function representation of an LTI system given by a state-space realization $(A,B,C,D)$.

   A = randn(6); B = randn(6,2); C = randn(3,6); D = zeros(3,2);
   sys = ss(A,B,C,D)

   Output:

   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)

   Output:

   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)

   Output:

   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}

Seminar 4. - Controllability, observability

Matlab examples

1. Compute the controllable and unobservable subspace of LTI state-space model $(A,B,C,D)$.

   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)

   Output:

   Cn =
        0    -4    -8
        0     0     0
        2     2    -2
   
   ans = 2

   Ctrb_Subsp = orth(Cn)

   Output:

   Ctrb_Subsp =
       0.9933   -0.1153
            0         0
       0.1153    0.9933

   Ctrb_Subsp = orth(sym(Cn))

   Output:

   Ctrb_Subsp =
   [ 0, -1]
   [ 0,  0]
   [ 1,  0]

   On = obsv(A,C), rank(On)

   Output:

   On =
        1     0     0
        1     2    -2
       -1     4    -4
   
   ans = 2

   Unobsv_Subsp = null(On)

   Output:

   Unobsv_Subsp =
            0
       0.7071
       0.7071

   Unobsv_Subsp = null(sym(On))

   Output:

   Unobsv_Subsp =
    0
    1
    1

Published Matlab scripts

1. Manipulations with Matlab's Symbolic Math Toolbox: d2017_09_14_gyak1. Basic symbolic routines, Laplace transformation, inverse Laplace transformation, partial fractional decomposition.
2. State-space transformations: d2017_09_19_gyak2.
   Dynamic systems simulations:
   • d2017_09_20_Rossler_attractor
   • d2017_09_20_Lorentz_system
   • d2017_09_20_brussellator
   • d2017_09_21_ccs_gyak2_eloben
   • Approximate solution of the heat equation with ode45 by considering the discretization along the spacial coordinate. (See also: solution using pdepe.)
   • Example for a discrete-time dynamical system: d2018_01_20_ex1_sim
3. Analysis in frequency/operator domain
   • d2017_09_27_gyak3_freqdom
   • d2017_09_28_gyak3_eloben
   • d2017_10_25_gyak6_PID_live
   • Advanced frequency domain analysis: minimum_phase_systems.
   • System inversion of minimum phase SISO systems.
4. Controllability, observability, subspaces: d2017_10_02_gyak4_subspaces.
   Geometrical meaning, staircase forms: d2017_10_04_gyak4_ctrb_obsb_geometrical_meaning.
   Fundamental theorem of the linear argebra, demonstration: d2017_10_05_gyak4_basicthm_LA.
5. Lyapunov function demonstration for a non-proper Lyapunov function (Hamiltonian system): d2017_10_11_gyak5_Lyapunov_improper.
   Lyapunov function demonstration for the pendulum system: d2017_10_11_gyak5_Lyapunov_pendulum_plain_vid
6. Matlab solutions of former homeworks (2017 fall)
   • Homework #1
   • Homework #2

Control systems demonstrations

Inverted pendulum model (inverz inga modell)

1. Model description and task sheet for the first Matlab practice: sem_08_2019_matlab1.pdf
2. Framework for the first Matlab practice
3. Model description and task sheet for the first Matlab practice: sem_10_2019_matlab2.pdf
4. Framework for the second Matlab practice
5. Model linearization around stable and unstable equilibrium point, simulation and analysis.
6. Inverted pendulum control and integral reference tracking.
7. Inverted pendulum control and integral reference tracking (augmented, corrected).
8. Inverted pendulum nonlinear control - 2018.07.30. (július 30, hétfő), 11:02
9. Advanced Nonlinear Control Methods (PhD course): Inverted pendulum linearization (Symbolic Math Toolbox) and pole placement
10. 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

Recommended literature

• G. Szederkényi, K. M. Hangos, J. Bokor: Computer Controlled Systems (CCS)
• Karl J. Åström and Richard M. Murray: Feedback Systems: An Introduction for Scientists and Engineers (FSB)

Additionally:

Andrew_Lewis_A_Mathematical_Approach_to_Classical_Control.pdf		2181 days		9.32 MB
Scherer.Weiland_2000_Linear_Matrix_Inequalities_in_Control.pdf		2181 days		1.25 MB
The_Fundamental_Theorem_of_Linear_Algebra.pdf		2181 days		330.57 KB

Extra material for the midterm tests

Former midterm examples
2014_ccs_1zh_A_feladatsor.pdf		2181 days		213.5 KB
2014_ccs_1zh_B_feladatsor.pdf		2181 days		201.93 KB
2015_ccs_1zh_elm.pdf		2181 days		111.9 KB
2015_ccs_1zh_gyak_MO.pdf		2181 days		124.17 KB
2015_ccs_2zh_elm.pdf		2181 days		104.75 KB
2015_ccs_2zh_gyak_MO.pdf		2181 days		114.2 KB
2015_ccs_gyakorlo_zh.pdf		2181 days		104.09 KB
2016_ccs_1zh_elm.pdf		2181 days		97.13 KB
2016_ccs_1zh_gyak.pdf		2181 days		84.4 KB
2016_ccs_2zh_elm.pdf		2181 days		83.64 KB
2016_ccs_2zh_gyak.pdf		2181 days		103.98 KB
2017a_ccs_1zh_computational.pdf		2181 days		87.01 KB
2017a_ccs_1zh_theoretical.pdf		2181 days		111.83 KB
2017a_ccs_2zh_elm.pdf		2181 days		104.28 KB
2017a_ccs_2zh_gyak.pdf		2181 days		131.4 KB
2017b_ccs_1zh_elm.pdf		2181 days		76.62 KB
2017b_ccs_1zh_elm_SOL.pdf		2181 days		136.01 KB
2017b_ccs_1zh_gyak.pdf		2181 days		90.72 KB
2017b_ccs_2zh.pdf		2181 days		162.78 KB
2017b_ccs_2zh_elm.pdf		2181 days		62.41 KB
2017b_ccs_2zh_gyak.pdf		2181 days		87.26 KB
2018_ccs_zh1_elm.pdf		2219 days		104.3 KB
2018_ccs_zh1_gyak.pdf		2219 days		107.2 KB
2018_ccs_zh2_elm.pdf		2177 days		90.57 KB
2018_ccs_zh2_gyak.pdf		2177 days		163.59 KB
2019_ccs_zh1_elm.pdf		1854 days		108.05 KB
2019_ccs_zh1_gyak.pdf		1857 days		112.16 KB
2019_ccs_zh1_gyak_MO.pdf		1855 days		175.28 KB
2019_ccs_zh1_gyak_TG.pdf		1854 days		122.07 KB
2019_ccs_zh1_gyak_TG_MO.pdf		1857 days		124.27 KB
Additional material
kawski_Lyapunov1.pdf		2181 days		481.87 KB
kawski_Lyapunov2.pdf		2181 days		262.33 KB

Extra material for the supplementary midterm

2015_ccs_potzh_elm.pdf		2181 days		101.32 KB
2015_ccs_potzh_gyak_MO.pdf		2181 days		117.14 KB
2016_ccs_potzh_elm.pdf		2181 days		96.75 KB
2016_ccs_potzh_elm_en.pdf		1808 days		100.7 KB
2016_ccs_potzh_gyak.pdf		2181 days		82.16 KB
2016_ccs_potzh_gyak_en.pdf		1808 days		84.95 KB
2017a_ccs_potzh.pdf		2181 days		139.12 KB
2017b_ccs_potpotzh .pdf		2181 days		117.83 KB
2017b_ccs_potzh.pdf		2181 days		85.86 KB
2018_ccs_potzh_elm.pdf		2163 days		91.27 KB
2018_ccs_potzh_gyak.pdf		2163 days		109.1 KB

Computer controlled systems (2018)

Lecture notes of the seminar

ccs_gyak02.pdf		2129 days		396.26 KB
ccs_hf3.pdf		1833 days		271.82 KB
Homeworks
hw_1_2018.pdf		2181 days		184.01 KB
hw_1_2018_sol_VM.pdf		2181 days		151 KB
hw_2_2018.pdf		2181 days		203.32 KB
hw_2_2018_sol_VM.pdf		2181 days		143.9 KB
hw_3_2018.pdf		2181 days		200.6 KB
hw_4_2018.pdf		2180 days		192.44 KB
hw_x_2018.pdf		2181 days		232.88 KB
hw_x_2018_extended_deadline.pdf		2181 days		232.88 KB
Matlab packages for TP students
lab_02_2018_TP_2018_09_18.zip		1473 days		323.69 KB
lab_02_2018_TP_exercises_for_Matlab_practice.pdf		2181 days		347.61 KB
lab_08_2018_TP_2018_11_06.zip		1473 days		1.75 MB
Lecture slides
lec_01_2018.pdf		2181 days		4.53 MB
lec_02_2018.pdf		2181 days		463.09 KB
lec_03_2018.pdf		2181 days		906.03 KB
lec_04_2018.pdf		2181 days		405.57 KB
lec_05_2018.pdf		2181 days		1017.58 KB
lec_06_2018.pdf		2181 days		991.21 KB
lec_07_2018.pdf		2181 days		1.63 MB
lec_08_2018.pdf		2181 days		1.09 MB
lec_09_2018.pdf		2181 days		1.03 MB
Seminars
sem_01_2018.pdf		2181 days		444.77 KB
sem_02_2018.pdf		2181 days		386.85 KB
sem_03_2018.pdf		2181 days		441.7 KB
sem_04_2018.pdf		2181 days		540.02 KB
sem_05_2018.pdf		2181 days		459.25 KB
sem_06_2018.pdf		2181 days		399.38 KB
sem_08_2018_matlab1.pdf		2181 days		311.63 KB
sem_09_2018.pdf		2181 days		210.95 KB
sem_10_2018_matlab2.pdf		2181 days		493.54 KB
sem_11_2018.pdf		2175 days		234.97 KB
Extra material and tutorial
xtra_material_Convex_optimization.pdf		2181 days		370.89 KB
xtra_material_DC_motor_model.pdf		2181 days		302.86 KB
xtra_material_Dissipativity_L2gain_LTI_LPV.pdf		2181 days		316.85 KB
xtra_material_Dissipativity_L2gain_LTI_LPV_dark.pdf		2181 days		3.02 MB
xtra_material_Laplace_table.pdf		2181 days		98.74 KB
xtra_material_lqservo.pdf		2181 days		30.01 KB
Model descriptions
xtra_model_crane.pdf		2181 days		217.4 KB
xtra_model_double_crane.pdf		2181 days		318.02 KB
xtra_model_gantry_crane_Ospina_Henao.pdf		2181 days		686.56 KB

Extra material

Former midterm examples
2014_ccs_1zh_A_feladatsor.pdf		2181 days		213.5 KB
2014_ccs_1zh_B_feladatsor.pdf		2181 days		201.93 KB
2015_ccs_1zh_elm.pdf		2181 days		111.9 KB
2015_ccs_1zh_gyak_MO.pdf		2181 days		124.17 KB
2015_ccs_2zh_elm.pdf		2181 days		104.75 KB
2015_ccs_2zh_gyak_MO.pdf		2181 days		114.2 KB
2015_ccs_gyakorlo_zh.pdf		2181 days		104.09 KB
2016_ccs_1zh_elm.pdf		2181 days		97.13 KB
2016_ccs_1zh_gyak.pdf		2181 days		84.4 KB
2016_ccs_2zh_elm.pdf		2181 days		83.64 KB
2016_ccs_2zh_gyak.pdf		2181 days		103.98 KB
2017a_ccs_1zh_computational.pdf		2181 days		87.01 KB
2017a_ccs_1zh_theoretical.pdf		2181 days		111.83 KB
2017a_ccs_2zh_elm.pdf		2181 days		104.28 KB
2017a_ccs_2zh_gyak.pdf		2181 days		131.4 KB
2017b_ccs_1zh_elm.pdf		2181 days		76.62 KB
2017b_ccs_1zh_elm_SOL.pdf		2181 days		136.01 KB
2017b_ccs_1zh_gyak.pdf		2181 days		90.72 KB
2017b_ccs_2zh.pdf		2181 days		162.78 KB
2017b_ccs_2zh_elm.pdf		2181 days		62.41 KB
2017b_ccs_2zh_gyak.pdf		2181 days		87.26 KB
2018_ccs_zh1_elm.pdf		2219 days		104.3 KB
2018_ccs_zh1_gyak.pdf		2219 days		107.2 KB
2018_ccs_zh2_elm.pdf		2177 days		90.57 KB
2018_ccs_zh2_gyak.pdf		2177 days		163.59 KB
2019_ccs_zh1_elm.pdf		1854 days		108.05 KB
2019_ccs_zh1_gyak.pdf		1857 days		112.16 KB
2019_ccs_zh1_gyak_MO.pdf		1855 days		175.28 KB
2019_ccs_zh1_gyak_TG.pdf		1854 days		122.07 KB
2019_ccs_zh1_gyak_TG_MO.pdf		1857 days		124.27 KB
Additional material
kawski_Lyapunov1.pdf		2181 days		481.87 KB
kawski_Lyapunov2.pdf		2181 days		262.33 KB

TP Seminar 1. (2018-09-11)

Basic linear algebra, Labplace transformation, solution of ODEs with Laplace transformation.

TP Seminar 2. (2018-09-18), Matlab practice

Sources:

• List of exercises for Matlab practice: ccs_lec02_TP_exercises_for_Matlab_practice.pdf
• Package for Matlab practice: ccs_lec2_TP_2018_09_18.zip
• Package after the Matlab practice (what we have done): ccs_lec2_TP_2018_09_18_after.zip

Solved Matlab exercises:

1. Basic linear algebra operations
2. Dynamic systems' simulation in Matlab
3. Also consider: approximate solution of the heat equation with ode45 by considering the discretization along the spacial coordinate. (See also: solution using pdepe.)

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)
• poly - Polynomial with specified roots or characteristic polynomial
• polyval - Polynomial evaluation
• roots - Polynomial roots
• w = conv(u,v) - Convolution and polynomial multiplication
• [q,r] = deconv(u,v) - Deconvolution and polynomial division

TP Seminar 3. (2018-09-25), Convex optimization

Convex optimization, primal-dual problem, introductio to LMIs.

Recommended literature: Scherer.Weiland_2000_Linear_Matrix_Inequalities_in_Control.pdf

Handwritten lecture notes: dark, original

TP Seminar 4. (2018-10-02)

System inversion of minimum phase SISO systems

TP Seminar 5. (2018-10-09)

Observability, controllability, model transformation into/from different model representations. Controllability staircase form.

TP Seminar 6. (2018-10-16), Dissipativity, stability

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

TP Seminar 8. (2018-11-06), Controllability and observability with LMIs

Tasks:

1. Compute controllability and observability staircase form for LTI systems and check their stabilisability and detectability.
2. Check stabilisability and detectability of LTI and affine LPV systems with LMIs (and design state feedback gain K and observer gain L).
3. Estimate the induced L2 operator norm of LTI and affine LPV systems using LMIs.
4. Design an optimal state feedback gain for LIT and affine LPV systems that minimizes the induced L2 operator norm.

Package for Matlab practice: lab_08_2018_TP_2018_11_06.zip

Things to do:

1. Extract the zip file.
2. Run Matlab in the root folder (ccs_lec8_2018_11_06) of the zip file, or navigate to the given directory in Matlab and run the startup script.
3. To test your environment run yalmiptest.

Published Matlab scripts:

1. Check if stabilisable (Success)
2. Check if stabilisable (Infeasible)
3. Affine LPV system computations with LMIs

Published Matlab scripts

1. Manipulations with Matlab's Symbolic Math Toolbox: d2017_09_14_gyak1. Basic symbolic routines, Laplace transformation, inverse Laplace transformation, partial fractional decomposition.
2. State-space transformations: d2017_09_19_gyak2.
   Dynamic systems simulations:
   • d2017_09_20_Rossler_attractor
   • d2017_09_20_Lorentz_system
   • d2017_09_20_brussellator
   • d2017_09_21_ccs_gyak2_eloben
   • Approximate solution of the heat equation with ode45 by considering the discretization along the spacial coordinate. (See also: solution using pdepe.)
   • Example for a discrete-time dynamical system: d2018_01_20_ex1_sim
3. Analysis in frequency/operator domain
   • d2017_09_27_gyak3_freqdom
   • d2017_09_28_gyak3_eloben
   • d2017_10_25_gyak6_PID_live
   • Advanced frequency domain analysis: minimum_phase_systems.
   • System inversion of minimum phase SISO systems.
4. Controllability, observability, subspaces: d2017_10_02_gyak4_subspaces.
   Geometrical meaning, staircase forms: d2017_10_04_gyak4_ctrb_obsb_geometrical_meaning.
   Fundamental theorem of the linear argebra, demonstration: d2017_10_05_gyak4_basicthm_LA.
5. Lyapunov function demonstration for a non-proper Lyapunov function (Hamiltonian system): d2017_10_11_gyak5_Lyapunov_improper.
   Lyapunov function demonstration for the pendulum system: d2017_10_11_gyak5_Lyapunov_pendulum_plain_vid
6. Matlab solutions of homeworks (2017 fall)
   • Homework #1
   • Homework #2

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

Computer controlled systems (2017 fall)

Eredmények

Events (midterms)

• October 19th - 1st midterm
• December 7th - 2nd midterm

Lecture notes of the seminar

Midterm sample test sheets for the 2nd midterm
2015_ccs_2zh_elm.pdf		2181 days		104.75 KB
2015_ccs_2zh_gyak_MO.pdf		2181 days		114.2 KB
2015_ccs_gyakorlo_zh.pdf		2181 days		104.09 KB
2015_ccs_potzh_gyak_MO.pdf		2181 days		117.14 KB
2016_ccs_2zh_elm.pdf		2181 days		83.64 KB
2016_ccs_2zh_gyak.pdf		2181 days		103.98 KB
2017_ccs_2zh_elm.pdf		2181 days		104.28 KB
2017_ccs_2zh_gyak.pdf		2181 days		131.4 KB
Lecture notes
ccs_gyak01.pdf		2181 days		311.19 KB
ccs_gyak02.pdf		2181 days		407.6 KB
ccs_gyak02_eng.pdf		2181 days		261.36 KB
ccs_gyak03.pdf		2181 days		441.73 KB
ccs_gyak04.pdf		2181 days		539.06 KB
ccs_gyak05.pdf		2181 days		459.68 KB
ccs_gyak06.pdf		2181 days		399.69 KB
ccs_gyak07.pdf		2181 days		260.77 KB
ccs_gyak08_matlabgyak1.pdf		2181 days		184.64 KB
ccs_gyak09.pdf		2181 days		150.44 KB
ccs_gyak10_matlabgyak2.pdf		2181 days		390.06 KB
ccs_gyak11.pdf		2181 days		181.51 KB
ccs_gyakorlo.pdf		2181 days		114.22 KB
Homework exercises
ccs_hf1.pdf		2181 days		351.62 KB
ccs_hf1_mo.pdf		2181 days		371 KB
ccs_hf2.pdf		2181 days		239.88 KB
ccs_hf2_mo.pdf		2181 days		289.16 KB
ccs_hf3 másolata.pdf		2181 days		224.15 KB
ccs_hf3.pdf		2181 days		224.15 KB
ccs_hf3_mo.pdf		2181 days		231.13 KB
ccs_hf4.pdf		2181 days		94.95 KB
ccs_lec01.pdf		2181 days		445.89 KB
ccs_lec02.pdf		2181 days		263.84 KB
ccs_lec02_v2.pdf		2181 days		261.76 KB
Model descriptions
ccs_model_crane.pdf		2181 days		217.4 KB
ccs_model_double_crane.pdf		2181 days		318.02 KB
ccs_model_gantry_crane_Ospina_Henao.pdf		2181 days		686.56 KB
ccs_segedlet_Matrixok.pdf		2181 days		141.88 KB
Extra material and tutorial
segedlet_Laplace_table.pdf		2181 days		98.74 KB
segedlet_szorgalmi_lqservo.pdf		2181 days		30.01 KB
midterm seating
zh1_beosztas.pdf		2181 days		44.72 KB
zh2_beosztas.pdf		2181 days		44.71 KB

Computer controlled systems (2017 spring)

week. Preliminaries

week. Laplace transformation

gyak2_Laplace

week. System representations

• Matlab script (system representation, transfer between representations, Bode diagram, controllability, observability): gyak3_sysrep_ctrb_obsb

week. Controllability, observability

• gyak4_ctrb_obsb_geometrical_meaning

week. BIBO and Lyapunov stability

• gyak5_BIBO