{"id":7560,"date":"2022-04-09T12:33:09","date_gmt":"2022-04-09T07:03:09","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=7560"},"modified":"2025-06-02T20:52:02","modified_gmt":"2025-06-02T15:22:02","slug":"what-is-the-state-transition-matrix","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-the-state-transition-matrix\/","title":{"rendered":"What is the State Transition matrix and its properties?"},"content":{"rendered":"\n<p>The State transition matrix is that matrix whose product with the state vector at initial time gives the value of variable x for time t. The state transition matrix is helpful for finding controllability, general solution, observability and stability of the LTI system.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">State Transition Matrix and its Properties:<\/h2>\n\n\n\n<p>As we know the output equation is given as<\/p>\n\n\n\n<p>y(t)=Cx(t)+Du(t)<\/p>\n\n\n\n<p>x(t)=Ax(t)+Bu(t)<\/p>\n\n\n\n<p>&nbsp;d(x)\/dt -Ax(t)=Bu(t)<\/p>\n\n\n\n<p>Taking Laplace transform of above equation we have<\/p>\n\n\n\n<p>SX(s)-X(0)-AX(s)=BU(s)<\/p>\n\n\n\n<p>SX(s)-AX(s)=BU(s)+X(0)<\/p>\n\n\n\n<p>[SI-A]X(s)=X(0)+BU(s)<\/p>\n\n\n\n<p>X(s)=[SI-A]<sup>-1<\/sup>[X(0)+BU(s)]<\/p>\n\n\n\n<p>X(s)=[SI-A]<sup>-1<\/sup>X(0)+[SI-A]<sup>-1<\/sup>BU(s) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (a)<\/p>\n\n\n\n<p>This is solution of state differential equation<\/p>\n\n\n\n<p>L<sup>-1<\/sup>X(s)= L<sup>-1<\/sup>{[SI-A]<sup>-1<\/sup>X(0)+[SI-A]<sup>-1<\/sup>BU(s)}<\/p>\n\n\n\n<p>x(t)= [SI-A]<sup>-1<\/sup>x(0)+ [SI-A]<sup>-1<\/sup>Bu(t)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (b)<\/p>\n\n\n\n<p>From above x(t) we can find output equation by replacing x(t) in output equation by its value from above equation(b)<\/p>\n\n\n\n<p>For the given system below<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"624\" height=\"201\" src=\"https:\/\/lh4.googleusercontent.com\/TbSOtpCuINdSWSVWfOQbkpiad8IEYKsZI9MLMSQhkHa3cTLibGYUE3D4DB19dWFVa9BBUU7_QKufkoHAzuHte9tuGjii2FOn3jM008JV2HnJGz6FlY_USGo2LmC1aLnkU3QnpacK\"><\/p>\n\n\n\n<p>The <a href=\"https:\/\/en.wikipedia.org\/wiki\/Transfer_function\">transfer function<\/a> of above system is given as<\/p>\n\n\n\n<p>TF=c(t)\/r(t)<\/p>\n\n\n\n<p>This c(t) is the output of the present system, which is not equal to above y(t), as their initial conditions are not considered.<\/p>\n\n\n\n<p>&nbsp;If initial conditions are zero than both y(t) and c(t) will be equal to<\/p>\n\n\n\n<p>L<sup>-1<\/sup>[SI-A]<sup>-1<\/sup>=\u03c6(t)\u2026\u2026.. <strong>state transition matrix<\/strong><\/p>\n\n\n\n<p>\u03a6(s)=[SI-A]<sup>-1<\/sup><\/p>\n\n\n\n<p>x(t)=\u03c6(t)x(0)+L<sup>-1<\/sup>[\u03c6(s)* BU(s)]&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <\/p>\n\n\n\n<p>When the input is zero i.e. u(t) =0 the State transition matrix satisfies the solution of the state equation&nbsp;<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"172\" height=\"28\" src=\"https:\/\/lh4.googleusercontent.com\/J8C9kcgFDPFFA7K6KTBCVTb6gY15sExNb9RyDczHG0YEwt5gg72tnpYmsHWxDI1Boq3i-SxilkApxZFJ8wPiB3egjXKu6zh2mIBeYbJL48ZZKQyRLoFNszZ2h6_mub6skxfBBZzC\"><\/p>\n\n\n\n<p>As u(t)=0<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"133\" height=\"87\" src=\"https:\/\/lh3.googleusercontent.com\/bO2RHJtrLHcfLS8x94n2Ckc3u0WV_et23D4VRhiuj1UYnUYc6TlGC3_I3BzA-D65G3zKtnFREyLOxveqvt6y0Ae9V38gid6f4E_na0GsLoBV8fLLxgBfVQ_ruGEKgRDbQeY7LbSR\"><\/p>\n\n\n\n<p>Solution to above equation is<\/p>\n\n\n\n<p>y(t)=Ke<sup>-Pt<\/sup>+e<sup>-Pt<\/sup>\u222b e<sup>Pt<\/sup> Q d(t)<\/p>\n\n\n\n<p>But [dx(t)\/dt] -Ax(t)=0<\/p>\n\n\n\n<p>Hence above equation becomes<\/p>\n\n\n\n<p>X(t)=Ae<sup>At<\/sup><\/p>\n\n\n\n<p>Substitute t=0<\/p>\n\n\n\n<p>x(0)=ke<sup>0<\/sup><\/p>\n\n\n\n<p>x(0)=k<\/p>\n\n\n\n<p>x(t)=x(0) e <sup>At&nbsp; &nbsp; &nbsp; &nbsp; <\/sup> &nbsp; &nbsp; &nbsp; &nbsp; (<strong>zero input response)<\/strong><\/p>\n\n\n\n<p>The above equation is the Zero Input Response of the system.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Properties<\/h2>\n\n\n\n<p>From above equation when u(t)=0<\/p>\n\n\n\n<p>x(t)=\u03c6(t) x(0)<\/p>\n\n\n\n<p>and from zero input response we have<\/p>\n\n\n\n<p>\u03c6(t)=e<sup>At<\/sup><\/p>\n\n\n\n<p><strong>Property 1:<\/strong><\/p>\n\n\n\n<p>\u03c6(0)= [I]<\/p>\n\n\n\n<p><strong>Property 2:<\/strong><\/p>\n\n\n\n<p>\u03a6<sup>-1<\/sup>(t)= [\u03c6(t)]<sup>-1<\/sup>=e<sup>-At<\/sup>=e<sup>A(-t)<\/sup><\/p>\n\n\n\n<p>\u03a6<sup>-1<\/sup>(t)= \u03a6(-t)<\/p>\n\n\n\n<p><strong>Property 3:<\/strong><\/p>\n\n\n\n<p>\u03a6<sup>K<\/sup>(t)= [\u03a6(t)]<sup>K<\/sup><\/p>\n\n\n\n<p>\u03a6<sup>K<\/sup>(t)=[e<sup>At<\/sup>]<sup>K<\/sup>=e<sup>A(tK)<\/sup><\/p>\n\n\n\n<p>Then  \u03a6<sup>K<\/sup>(t)= \u03a6(Kt)<\/p>\n\n\n\n<p><strong>Property 4:<\/strong><\/p>\n\n\n\n<p>\u03a6(t<sub>1<\/sub>+t<sub>2<\/sub>)=e<sup>A(t<\/sup><sub>1<\/sub><sup>+t<\/sup><sub>2<\/sub><sup>)<\/sup><\/p>\n\n\n\n<p><sup>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/sup><sup> <\/sup>=e<sup>(At<\/sup><sub>1<\/sub><sup>+At<\/sup><sub>2<\/sub><sup>)<\/sup>=e<sup>At<\/sup><sub>1<\/sub> * e<sup>At<\/sup><sub>2<\/sub><\/p>\n\n\n\n<p>\u03a6(t<sub>1<\/sub>+t<sub>2<\/sub>)= \u03a6(t<sub>1<\/sub>)\u03a6(t<sub>2<\/sub>)<\/p>\n\n\n\n<p><strong>Property 5:<\/strong><\/p>\n\n\n\n<p>\u03a6(t<sub>2<\/sub>-t<sub>1<\/sub>) * \u03c6(t<sub>1<\/sub>-t<sub>0<\/sub>)=e <sup>A(t<\/sup><sub>2<\/sub><sup>-t<\/sup><sub>1<\/sub><sup>)<\/sup> * e <sup>A(t<\/sup><sub>1<\/sub><sup>-t<\/sup><sub>0<\/sub><sup>)<\/sup><\/p>\n\n\n\n<p>\u03a6(t<sub>2<\/sub>-t<sub>1<\/sub>) * \u03c6(t<sub>1<\/sub>-t<sub>0<\/sub>)= \u03a6(t<sub>2<\/sub>-t<sub>0<\/sub>)<\/p>\n\n\n\n<p><strong>Example<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>If matrix <img loading=\"lazy\" decoding=\"async\" width=\"112\" height=\"60\" src=\"https:\/\/lh5.googleusercontent.com\/USpTgMTvcAMk7WlvoA7ilFZ1e3dhgSOTKitXJgYdl9mo13V_MPdXy20_QMY3AhSAhC7vzpl2SIwvOLRppy0phlQzYJ0RHaHh-9GcCI8xsTSYsPV4V00WhGut2fWAEqZ4SC_25XZH\"> &nbsp;Find the state transition matrix?<\/li><\/ol>\n\n\n\n<p>Solution: The state transition matrix is given by L<sup>-1<\/sup>[SI-A]<sup>-1<\/sup>=\u03c6(t)<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"321\" height=\"165\" src=\"https:\/\/lh3.googleusercontent.com\/jABkLoCKVb5EmZVYe7jbf1ZxjdQY5oGUUSksKfvMZjZ1DdBaG4EcCmDckO-bfxsjC89kR7_fUXtZOV_uIQRX3CWPlOwVTwDD9aWWVcyaWnWlU5FW-a_Tu_gmD-_uk8loTV0tCdt7\"><\/p>\n\n\n\n<p>Taking inverse Laplace of above, we get<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"357\" height=\"118\" src=\"https:\/\/lh6.googleusercontent.com\/kY3P52Aa_72J14ie-dhHjqAhkKzMxeKUNPzigrt9wM5OZnDZ-0X7yn-TZRhVetzKpm106X0RcDF7jNH_rh1MsfJ9v4x6p-uwsmR-Cp2cH-5SxJWRNF7VTQTh4_t0Uz91_ZvPtzdb\"><\/p>\n\n\n\n<p>Hence \u03c6(t)=L<sup>-1<\/sup>[SI-A]<sup>-1<\/sup><\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"225\" height=\"49\" src=\"https:\/\/lh3.googleusercontent.com\/yyQYYDp7td_xu68enNdVkgqv0_vKN17aUJDuH_1eiAFGqzubZZJiUs5vbMTTZantzeDDROs6ZUsLV7jVbvKcvuZwjSHwXmqXaSkej03UDLYhSBGTFo0SoFH4pltObd3O8Xq249Rg\"><\/p>\n\n\n\n<p>2) Find \u03c6(t) if <\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"131\" height=\"49\" src=\"https:\/\/lh6.googleusercontent.com\/GcOHhSddNhiuuwzkTdo4lhv454sXkD-7wUzAdwqXBlraqwBEf6c_vVlCZt3UgHv1vzhX7NyuHm2CKFowsKkZ7LTkTm3qnDWROBIjJp9nwzFNm0PIB_AoDADdWb80UZWbXqqIxsRm\"><\/p>\n\n\n\n<p>Solution: As we already know that L<sup>-1<\/sup>[SI-A]<sup>-1<\/sup>=\u03c6(t)<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"336\" height=\"174\" src=\"https:\/\/lh3.googleusercontent.com\/NbsBkafY6bd9ugDGHRapWAFvpHuL14LxjxDeX0XJBvgUIR4Ouj8FJKIWQevD2JqI8lbfS7CLhPg8CdxOwnt8YVngb8cwj0VtaoO2R1T4UC2alX61p2WTedqHN3qzgKJKXcdm9FNB\"><\/p>\n\n\n\n<p>Hence \u03c6(t)=L<sup>-1<\/sup>[SI-A]<sup>-1<\/sup><\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"190\" height=\"53\" src=\"https:\/\/lh4.googleusercontent.com\/GgyS5vL4J6WDqf35TFMiQiNs-ApHXffKrAuEsX_oU1JhHm5QA-an5bZuL3Epo4YP-4E-Y5LKKZ1CznFo9vRC1Gwg-I0A9gdiRZ4mIwTQ8d8AciSNCeXPU1Onl5uqu8xyFmPBjoDa\"><\/p>\n\n\n\n<p><strong>Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-zigbee\/\" target=\"_blank\" rel=\"noreferrer noopener\">What is ZigBee?<\/a><\/li><li><a href=\"https:\/\/www.goseeko.com\/blog\/what-are-encoders-and-decoders\/\" target=\"_blank\" rel=\"noreferrer noopener\">Encoders and Decoders<\/a><\/li><li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-uart\/\" target=\"_blank\" rel=\"noreferrer noopener\">Working of UART<\/a><\/li><li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-fft\/\" target=\"_blank\" rel=\"noreferrer noopener\">Classification of FFT<\/a><\/li><li><a rel=\"noreferrer noopener\" href=\"https:\/\/www.goseeko.com\/blog\/what-is-fourier-transform\/\" target=\"_blank\">Fourier Transform and its properties<\/a><\/li><\/ul>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The State transition matrix is that matrix whose product with the state vector at initial time gives the value of variable x for time t. <\/p>\n","protected":false},"author":3,"featured_media":7562,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[37],"tags":[],"class_list":["post-7560","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-electronics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.3.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is the State Transition matrix and its properties? 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