How to build a discrete model of my continous model. The command c2d does not work properly.

Hi,

i need help at the conversion from a continuous model to a discrete form with the c2d command. I have a 60th order continuous transfer function which describes a mechanical model. For simulations in simulink i and want to use a discrete form of it.

Unfortunately the result of the c2d command differs strongly from the bode diagram of the original form (see attached pictures). I have tried all methods of the c2d-command without success. I assume there might be a numerical issue, because some of the coefficients (of the original tf) are very huge. Is someone out there who can help me :)

I would be very grateful for an example how to transfer the model into a discrete form. Thank you for your support!

To make it easier for you helpers i put the code directly into this post. So you can copy it directly without the need to download additional files.

num = [0 0.0109927448859302 3.99448177501850 109517.592486183 37649448.3297623 507611867137.250 164838549869325 1.45602535690666e+18 4.45881503809445e+20 2.90035083415998e+24 8.36041014603243e+26 4.26790523383173e+30 1.15569783404456e+33 4.81826502150825e+36 1.22295666280531e+39 4.27845058605768e+42 1.01539760485039e+45 3.03938186177652e+48 6.72647987429533e+50 1.74778155081593e+54 3.59612247521501e+56 8.20078794125948e+59 1.56346079593921e+62 3.15574619755332e+65 5.55367163321311e+67 9.98668712242453e+70 1.61548609147541e+73 2.60096532030733e+76 3.84889746257468e+78 5.56890076412877e+81 7.49763777876622e+83 9.77459134207665e+86 1.18988212041540e+89 1.40004341630049e+92 1.52999912233273e+94 1.62606485296444e+97 1.58210557703146e+99 1.51876715174348e+102 1.30296737344856e+104 1.12896104397887e+107 8.44300479163441e+108 6.59354200483496e+111 4.23987957489757e+113 2.97831644029417e+116 1.61935716620884e+118 1.02056128827901e+121 4.59433801637339e+122 2.58996115354035e+125 9.39264473695559e+126 4.72081548229835e+129 1.32761017997142e+131 5.92985806058965e+133 1.21955706185081e+135 4.82591650333776e+137 6.53675774784677e+138 2.28520008656776e+141 1.59942579434838e+142 4.92935182857493e+144 4.93641526749852e+144 1.34409420480861e+147 2.29227106147305e+133];
den = [1 370.263955149734 10046214.1163391 3521824165.73777 46983215434669.3 1.55712711040320e+16 1.36069375081261e+20 4.25659491601838e+22 2.73862539719748e+26 8.07225305323028e+28 4.07494720576437e+32 1.12956341470129e+35 4.65569456428359e+38 1.21110749550192e+41 4.18753577989324e+44 1.01989293621356e+47 3.01620740808921e+50 6.86017031860269e+52 1.76048222329015e+56 3.72853584744324e+58 8.39418194818009e+61 1.65017547779661e+64 3.28674574561815e+67 5.97598679808093e+69 1.05987254824076e+73 1.77518647761284e+75 2.81734716413742e+78 4.32722451099141e+80 6.16811900808935e+83 8.64312729787564e+85 1.10940180060664e+89 1.40999353485659e+91 1.63241782537416e+94 1.86921793202717e+96 1.95359139835591e+99 1.99986712397735e+101 1.88704196973634e+104 1.71144761109558e+106 1.45721168144069e+109 1.15844783506811e+111 8.89107686850098e+113 6.11633652831429e+115 4.22514277263861e+118 2.47562014649315e+120 1.53643864600351e+123 7.51534256603831e+124 4.18177325474872e+127 1.66297724028395e+129 8.27716813141826e+131 2.57876359985128e+133 1.14513572297659e+136 2.64141469207809e+137 1.04301772387348e+140 1.61407312523668e+141 5.64022929561847e+143 4.71923594864417e+144 1.44844295315995e+147 2.81630622346565e+147 7.52320360199863e+149 4.66283030891243e+149 1.09870911608763e+152];

Ts = 1/2048;

% Continuous System
sysc = tf(num,den);
bode(sysc);

% Discrete System
sysd = c2d(sysc,Ts);
figure;
bode(sysd);

frequency response of the continuous system

frequency response of the synthesized discrete form.

[Jan]

Du erreichst mehr Leser, wenn Du die in diesem Forum übliche sprache Englisch verwendest.

[Roman Kraus]

Thank you Jan for your suggestion to translate my request into english.

[Mischa Kim]

I recommend going a step further and changing the question title, as well, since most contributors will skip this post just for the non-English title.

[Roman Kraus]

Ok, you are absolutely right. I changed the title,too :D. Embarrassing.

Answer 1

[Arkadiy Turevskiy]

Transfer function representation of an LTI system is not the best representation for doing calculations. For more on that please see this example .

Specifically, for your problem, convert sysc to a state space representation, do c2d conversion on that state-space system.

sysc_ss=ss(sysc);
sysd_ss=c2d(sysc_ss,Ts);;

P = bodeoptions; % 
P.PhaseWrapping = 'on'; % turn phase wrapping on

bodeplot(sysc,sysd_ss,P);
legend('cont. tf', 'discrete ss')

This looks much better :)

There are a couple of additional things to try:

If you want to match the frequency response, the best discretization method to use is Tustin .

Also, the plot above shows that it would be nice to improve accuracy in the frequency range from 100 rad/s to 1,000 rad/s. This can be done using prescale command.

So, further improving the code:

sysd_ss=c2d(sysc_ss,Ts,'Tustin');
sysd_ss_p=prescale(sysd_ss,{100 1000});
figure;
bodeplot(sysc,sysd_ss_p,P);
legend('cont. tf', 'discrete ss with Tustin and scaling')

[Arkadiy Turevskiy]

You can try improving thing further by using pre-wapring with Tustin.

sysc_ss=ss(sysc);

opt = c2dOptions('Method', 'tustin', 'PrewarpFrequency', 500);

sysd_ss=c2d(sysc_ss,Ts,opt);

sysd_ss_p=prescale(sysd_ss,{100 1000});

bodeplot(sysc,sysd_ss_p,P);

You can then play with the prewarping frequency value to see which one gives you the best fit.

[Roman Kraus]

Hello Arkadiy,

thank you a lot for your answer which seems to be exactly what I needed! A first simple test with matlab and with simulink gave me very promising results.

Originally the tf-model results from a state-space model, created by ss2tf. I tried to create a discrete ss model with ss(A,B,C,D,Ts) without good results (it seemed the ss-representation was still a continuous one, the ss-help gives some explanation), so i tried to simplify the system by building a transfer function. Now I know i should stay at the state-space form and should convert it with c2d.

Thank you for your investment in time and the helpful code.

Kind regards from Frankfurt,

Roman