# Confusion regarding stability & step response?

Let's say we have a system (for example an integrator) whose step response has a negative slope as shown by the MATLAB code below:

clc;clear ;close all
R=47*10^3
C=1*10^-7
num1=[-1]
den=[R*C 0]
sys1=tf(num1,den)
num2=
sys2=tf(num2,den)
subplot 211
pzmap(sys1)
title('pole zero map for pure integrator, with -1 in numerator')
subplot 212
pzmap(sys2)%plotting poles and zeros
title('pole zero map for integrator, with +1 in numerator')
figure
subplot 211
step(sys1)
title('step response for pure integrator, with -1 in numerator')
subplot 212
step(sys2)
title('step response for integrator, with +1 in numerator')


sys1 is an ideal integrator and we know that ideal integrators are stable systems.

But what if we replace -1 in the numerator with 1 (sys2): then the step response slope will be positive: is sys2 unstable because of the positive slope?

• What do you mean by "pure" integrator, an ideal integrator? Anyway, changing the sign of a transfer function will definitely not change its stability properties. Sep 7, 2022 at 7:47
• An ideal integrator with a transfer function of $1/s$ is only marginally stable since it has pole at $s=0$ and infinite gain at DC. That's why the step response of the integrator grow to inifinity Sep 7, 2022 at 11:12
• @MattL. please check updated question Sep 7, 2022 at 11:20
{"ft_lang_label":"__label__en","ft_lang_prob":0.6923682,"math_prob":0.95755816,"size":851,"snap":"2023-40-2023-50","text_gpt3_token_len":244,"char_repetition_ratio":0.1475797,"word_repetition_ratio":0.11111111,"special_character_ratio":0.2867215,"punctuation_ratio":0.077380955,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98229617,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-29T11:01:02Z\",\"WARC-Record-ID\":\"<urn:uuid:5701cc5d-bcc2-4a54-8442-2b83aab56afa>\",\"Content-Length\":\"163753\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:879991e7-4b06-4356-9107-8f5da15f1849>\",\"WARC-Concurrent-To\":\"<urn:uuid:ce0eb73d-22ef-42ca-a66e-c4e10dc0edcb>\",\"WARC-IP-Address\":\"104.18.10.86\",\"WARC-Target-URI\":\"https://dsp.stackexchange.com/questions/84457/confusion-regarding-stability-step-response\",\"WARC-Payload-Digest\":\"sha1:GUJUAHM3VRADQOKF5JI3DUXDCUECR5NJ\",\"WARC-Block-Digest\":\"sha1:NMS675F23WIW72KEPMSEZSAUI7NNOWTI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510501.83_warc_CC-MAIN-20230929090526-20230929120526-00476.warc.gz\"}"}