The model order is equal to the order of the denominator polynomial. Below is the step response of the system computed by using the MATLAB function step (). "itemListElement": Which shows that G(s) has a single zero at s = -0.75, two poles at s = -1 and -5, and a gain of 4. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Matlab Transfer Function Operations with Examples By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. }{\mathop{u}}\,+7u\], Taking the Laplace transform of the differential equation with zero initial conditions, we obtain the polynomial form of the transfer function as, \[G(s)=\frac{3{{s}^{2}}+5s+3}{{{s}^{4}}+3{{s}^{3}}+4{{s}^{2}}+2.5s+0.8}\]. Find centralized, trusted content and collaborate around the technologies you use most. The two forms of the command that are applicable to transfer-function models are pzmap(numG,denG) and pzmap(pG, zG), where in the rst case the arguments are row vectors and in the latter case, the arguments are column vectors. Not the answer you're looking for? that method just gives H(s) still. Did you find apk for android? 1 ----- s + 8 Continuous-time transfer function. delay) in the time domain is interpreted as complex phase shifts in the frequency domain. How can get transfer function. - MathWorks As the result, from (1), we obtain. "@context": "http://schema.org", We can pass one, two, or three arguments through ztrans function. You can also select a web site from the following list: Select the China site (in Chinese or English) for best site performance. Check the example below. Fourier transform - Wikipedia For instance, consider a continuous-time SISO dynamic system represented by the transfer function sys (s) = N (s)/D (s), where s = jw and N (s) and D (s) are called the numerator and denominator polynomials, respectively. transferFunction=tf (system1) We define the state-space model by using the MATLAB function ss (). There are two approaches. }{\mathop{u}}\,+3u & \cdots & (1) \\\end{matrix}$. Based on "@id": "https://electricalacademia.com", "@type": "ListItem", "name": "Transfer Function Calculation Matlab with Example" transfer function in matlab code For the Ackerberg-Mossberg with a summing circuit the Transfer Function is the following equation below. where is the transfer function and is the complex variable (obtained by applying the Laplace transform), and and are the Laplace transforms of the output and input vectors. Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide, matlab-create a matrix of transfer function, Making location easier for developers with new data primitives, Stop requiring only one assertion per unit test: Multiple assertions are fine, Mobile app infrastructure being decommissioned. Similarly, given a SYS object created from a transfer function, a state-space model of the system can be found using "ss (sys)". Reload the page to see its updated state. "name": "Control with Matlab" Transfer Function Models - MATLAB & Simulink - MathWorks Italia How can I define an n-by-n transfer function matrix in matlab? Mechanical Engineering questions and answers. The zpk model object represents transfer functions in factorized form. what if i wanted to write an equation that was a function of 's+x', where x is a constant. Here is the first approach that relies upon the MATLAB Symbolic Toolbox. Why is SQL Server setup recommending MAXDOP 8 here? This approach relies upon the MATLAB Control System Toolbox. From the first equation, we have: where is the identity matrix. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. MIMO transfer functions are arrays of SISO transfer functions. By substituting (5) in the output equation of (3), we obtain, The last equation can be written compactly, In the case of single-input-single-output systems, the transfer function is not a matrix, and we can write, Problem: Find the transfer function of the following system, where is the adjugate matrix of and is the determinant matrix of . Matlab Z Transform | Examples of Matlab Z Transform - EDUCBA Robot - Wikipedia You can also select a web site from the following list: Select the China site (in Chinese or English) for best site performance. Compare the response with that obtained with the step command applied . Other MathWorks country We define the state-space model by using the MATLAB function ss(). Why do I get two different answers for the current through the 47 k resistor when I do a source transformation? The result will be the three-tuple [zz, pp, kk] , which consists of the values of the zeros, poles, and gain of G(s), respectively. Transfer function 'tf'. I want each element to be of tf data type. (Matlab), Making a matrix of strings read in from file using feof function in matlab, design a Matlab filter from measured transfer function, Matlab create transfer function for given settling time and overshooting, Meaning of rational transfer function underlying MATLAB filter or Scipy.signal filter, Analysis of the stability of the system characterized by the transfer function - MATLAB. The use of the 'residue()' command is very simple. Unable to complete the action because of changes made to the page. A robot is a machineespecially one programmable by a computercapable of carrying out a complex series of actions automatically. The function file quadratic.m will contain the primary function quadratic and the sub-function disc, which calculates the discriminant. Determine the system function H (z), magnitude response H (ej), and phase response H (ej) of the following systems. determine the partial-fraction expansion of G(z). (7 points) With H (z) and h[n] Found in point a. find the system response to a step input x[n] =u[n] c. (5 points) Find . Definition of Matlab Z Transforms Z transform is used to convert discrete time signals into the frequency domain. To create the system described in (3), we would enter the commands zG = -0.75, pG = [-1; -5],and kG = 4. . Unable to complete the action because of changes made to the page. Discrete time signal includes real as well as imaginary values (Complex numbers). We explain how to perform this conversion analytically and in MATLAB. Here, we explain how to use the 'ss2tf()' command in Matlab with a very . The active use of " Matlab Function " expressions in the MATLAB /SIMULINK program especially for Script Application is one of the contents of this course. The cofactor matrix of is (see the video for the explanation on how to compute the cofactor matrix). Transfer Function, Frequency Response and Magnitude | Chegg.com Obtain a plot of the step response by adding a pole at s = 0 to G (s) and using the impulse command to plot the inverse Laplace transform. Accelerating the pace of engineering and science. Create a function file quadratic.m and type the following code in it function [x1,x2] = quadratic(a,b,c) %this function returns the roots of % a quadratic equation. Note that the Laplace transforms of vector are denoted by capital letters. 3 Methods of Transfer Function in Matlab - EDUCBA Find the transfer function in both the polynomial (TF) and factored (ZP) forms for the fourth-order system whose differential equation is, \[\overset{.}{\mathop{y}}\,+3\overset{}{\mathop{y}}\,+4\overset{..}{\mathop{y}}\,+2.5\overset{.}{\mathop{y}}\,+0.8y=3\overset{..}{\mathop{u}}\,+5\overset{. If we know the numerator and denominator polynomials of G(s), we can represent the model in MATLAB by creating a pair of row vectors containing the coefcients of the powers of s, in descending order, of the numerator and denominator polynomials. "item": I tried with the definition: MATRIX=zeros(n); but each element of MATRIX is of double type. "url": "https://electricalacademia.com/category/control-system-matlab/", Does activating the pump in a vacuum chamber produce movement of the air inside? In the above system, the input is x (t) and the output is y (t). Bode Plot Matlab | How to do Bode Plot Matlab with examples? - EDUCBA Transfer Function Calculation Matlab with Example [ Transfer Functions - MATLAB & Simulink - MathWorks MATLAB Program for finding Poles & Zeroes of Transfer Function How can I define an n-by-n transfer function matrix in matlab? offers. It is obtained by applying a Laplace transform to the differential equations describing system dynamics, assuming zero initial conditions. },{ I used the following code to find find fourier transfrom of gaussian curve. Whereas: ZC = 1 jC ZL = jL Z C = 1 j C Z L = j L. Now, let's compute the transfer function using Matlab: . { Are Githyanki under Nondetection all the time? If you are using an Academic license then it is possible that your institution has already arranged licenses for you; you would need to check with your IT people about that. Let us apply the Laplace transform to our original system (1). "url": "https://electricalacademia.com", General transfer function of the Akerberg-Mossberg with a summing circuit Now with this transfer function, we have 5 variables that we can play with to obtain our desired frequency response. here, i demo how to use matlab to get the transfer function of your overall block diagram, and to save yourself some tedious calculations.the last image of video shows the hand. Lab 4 : MATLAB for controls - state space analysis - UMD From the numerical output we nd that G (s) has zeros at s = -0.8333 j1.2802, poles at s == -.4548j0.5420 and s = -1.0452j0.7110, and a gain of 3. Our goal is to convert this state-space model to a transfer function (also called a transfer matrix, since the system has multiple inputs and outputs): the transfer function will have the following form. We and our partners use data for Personalised ads and content, ad and content measurement, audience insights and product development. Once we have the system model in either the TF or the ZP form, we can obtain a graphical representation of the transfer functions poles and zeros by using the pzmap command from the Control System Toolbox. I prefer women who cook good food, who speak three languages, and who go mountain hiking - what if it is a woman who only has one of the attributes? I would like to implement something that is in the form of H(s+5). The command syms is used to define a symbolic complex variable . Any helps with implementation would be great, that is not quite what i am looking for. How can I get a huge Saturn-like ringed moon in the sky? Transfer Function Represent linear time-invariant systems in the frequency domain A transfer function is a convenient way to represent a linear, time-invariant system in terms of its input-output relationship. MIMO transfer functions are arrays of SISO transfer functions. where is the Laplace transform of the state vector. https://www.mathworks.com/matlabcentral/answers/1598744-how-to-plot-transfer-functions, https://www.mathworks.com/matlabcentral/answers/1598744-how-to-plot-transfer-functions#comment_1859089, https://www.mathworks.com/matlabcentral/answers/1598744-how-to-plot-transfer-functions#answer_843264, https://www.mathworks.com/matlabcentral/answers/1598744-how-to-plot-transfer-functions#comment_1859274. Alternatively, if the zeros, poles, and gain are known as the column vectors zG and pG and the scalar kG, we would enter the command [num, den] = zp2tf (zG, pG, kG) . Transfer function models describe the relationship between the inputs and outputs of a system using a ratio of polynomials. RLC Circuit Transfer Function Calculation using Matlab "name": "Home" Choose a web site to get translated content where available and see local events and Do you want to see the resulting expression as a function of (s+x), or do you just want to write an equation as function of (s+x)? Did Dick Cheney run a death squad that killed Benazir Bhutto? Matlab Code to Find the Transfer Function },{ Transfer Function in MATLAB: As noted previously that the transfer function represents the input and output of the system in terms of the complex frequency variable so that the transfer function can give the complete information about the frequency response of the system. The command also depends on the number of inputs and outputs in your system. The documentation has good examples on how to write transfer functions: http://www.mathworks.com/help/toolbox/control/ref/tf.html. "position": 3, The Code [D,S,R] = xlsread ('data.csv'); FFT_y = fft (y); %take fourier transform FF_mag = abs (FFT_y )/ (length . Finally, our transfer function matrix is: Below are the MATLAB codes for computing the transfer function. Other MathWorks country Convert State-Space Models into Transfer Functions with MATLAB Codes The adjugate matrix is formed by transposing the cofactor matrix. Numerical integration techniques, transfer-function modeling, harmonic analysis, and power quality performance assessment; MATLAB/Simulink, Power Systems Toolbox, and PSIM for the simulation of power electronic circuits, including for renewable energy sources such as wind and solar sources If it is the latter you could do this: You may receive emails, depending on your. In Matlab ztrans function is used to find the z transform of any signal. >> h=1/ (arg+3) h = 1 ------ s + 13 Continuous-time transfer function. Presents numerical integration techniques, transfer-function modeling, harmonic analysis and power quality performance assessment Examines existing software such as, MATLAB/Simulink, Power Systems Toolbox and PSIM to simulate power electronic circuits including the use of renewable energy sources such as wind and solar sources First, the Fourier transform indicates that there are 3 poles, one at the origin, one at about 550 KHz, and one at infinity (actually the Nyquist frequency, that is for all practical purposes infinity), and 2 zeros, between each pair of the poles. "@id": "https://electricalacademia.com/control-system-matlab/transfer-function-calculation-matlab-with-example/", The transfer function can be determined by the following relation: H (f) = V o V 1 = ZC ZC+ZL+R H ( f) = V o V 1 = Z C Z C + Z L + R. A transfer function is simply a ratio between input and output. "position": 1, The consent submitted will only be used for data processing originating from this website. Stack Overflow for Teams is moving to its own domain! The other parts of the code are self-explanatory. How can we create psychedelic experiences for healthy people without drugs? The MATLAB commands in Script 1 will create the polynomial form of G (s) and then use it to determine the factored form. Use the pole-zero pattern to explain the shape of their magnitude responses. Transfer function in matlab code - vhiu.hunde-gourmet-bar.de your location, we recommend that you select: . The tf model object represents transfer functions in polynomial form. Advanced Transfer function model - MATLAB - MathWorks { Some of our partners may process your data as a part of their legitimate business interest without asking for consent. The roots of the denominator polynomial are referred to as the model poles. Transfer Function - MATLAB & Simulink - MathWorks Faster way to initialize arrays via empty matrix multiplication? The roots of the numerator polynomial are referred to as the model zeros. Then using two variables of transfer function 'num1' and 'den1', we can display the transfer function and stored it in variable 'TF1'.Then use the bode function in brackets the variable which is assigned for transfer function 'TF1'. From this equation, we have, where is an inverse of . How to plot transfer functions?. If the transfer function is known in terms of its zeros, poles, and gain, we can create the model by entering column vectors for the zeros and poles, and enter the gain as a scalar. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. You may receive emails, depending on your. Commands for Creating Transfer Functions From the numerical output we nd that G(s) has zeros at s = -0.8333 j1.2802, poles at s == -0.4548j0.5420 and s = -1.0452j0.7110, and a gain of 3. I just need to initialize it since I will load it with the correct tf in a second part of my code. Hi, I want to write a transfer function in matlab. } When a model has been described in MATLAB in either one of these forms, the Control System Toolbox provides several functions to obtain the description in the other form.if(typeof ez_ad_units != 'undefined'){ez_ad_units.push([[250,250],'electricalacademia_com-medrectangle-4','ezslot_2',142,'0','0'])};__ez_fad_position('div-gpt-ad-electricalacademia_com-medrectangle-4-0'); For example, if the numerator and denominator polynomials are known as the vectors numG and denG, we merely enter the MATLAB command [zz, pp, kk] = tf2zp (numG, denG). Asking for help, clarification, or responding to other answers. The transfer function of a system is given below Determines the poles and zeroes and show the pole-zero configuration in s-plane using MATLAB. "@type": "ListItem", Make a wide rectangle out of T-Pipes without loops. Non-anthropic, universal units of time for active SETI, Looking for RF electronics design references. please write a logical matlab code for the transfer function of this shape. Arkadiy Savan Prajapati on 24 Jul 2020 0 Link num = 100 den = [1,14,10] sys = tf (num,den) printsys (num,den) Sign in to answer this question. Start Hunting! MATLAB:Transfer Functions - PrattWiki - Duke University An easy way to create a transfer function in MATLAB involves using the command s=tf('s') to create s as a variable and then use s in a line of code to make a transfer function. For example: G(s)= [ (s3)/ (s+4) ; (s+1)/ (s+2)] is a one-input, two output transfer function. Transfer function 'tf' - MATLAB Answers - MATLAB Central - MathWorks Based on We and our partners use cookies to Store and/or access information on a device. where is the state vector, is the input vector, and is the input vector, the matrices , , and are the system matrices. Then we transform such a state-space model into a transfer function form by using the command tf(). Try these codes below please; clc; clear; close all; numerator = 1; denominator = [1,1,3,1]; . When a model has been described in MATLAB in either one of these forms, the Control System Toolbox provides several functions to obtain the description in the other form. Reload the page to see its updated state. That is all we need to estimate the transfer function. sites are not optimized for visits from your location. Book where a girl living with an older relative discovers she's a robot. 2022 Moderator Election Q&A Question Collection. MathWorks is the leading developer of mathematical computing software for engineers and scientists. >> arg=tf ('s')+10 arg = s + 10 Continuous-time transfer function. Write a MATLAB program to determine the partial fraction expansion of a rational z-transform. Using this program. https://www.mathworks.com/matlabcentral/answers/34274-writing-transfer-functions-in-matlab, https://www.mathworks.com/matlabcentral/answers/34274-writing-transfer-functions-in-matlab#comment_1956875, https://www.mathworks.com/matlabcentral/answers/34274-writing-transfer-functions-in-matlab#answer_42967, https://www.mathworks.com/matlabcentral/answers/34274-writing-transfer-functions-in-matlab#comment_71481, https://www.mathworks.com/matlabcentral/answers/34274-writing-transfer-functions-in-matlab#comment_71486, https://www.mathworks.com/matlabcentral/answers/34274-writing-transfer-functions-in-matlab#comment_71491, https://www.mathworks.com/matlabcentral/answers/34274-writing-transfer-functions-in-matlab#answer_43324, https://www.mathworks.com/matlabcentral/answers/34274-writing-transfer-functions-in-matlab#answer_470338. In factorized form polynomial form of SISO transfer functions are arrays of SISO functions! # x27 ; function & # x27 ; residue ( ) Server setup recommending MAXDOP here... Clear matlab transfer function code close all ; numerator = 1 -- -- s + 8 Continuous-time transfer models. The cofactor matrix of is ( see the video for the current through the 47 k when... Syms is used to define a Symbolic complex variable Control system Toolbox MATLAB function ss ( ) to to! Transforms of vector are denoted by capital letters are arrays of SISO transfer are. Experiences for healthy people without drugs, https: //www.educba.com/bode-plot-matlab/ '' > can. Explanation on how to do Bode Plot MATLAB with examples apply the Laplace transform our. ) matlab transfer function code the output is y ( t ) < /a > as the model.. Show the pole-zero configuration in s-plane using MATLAB primary function quadratic and the sub-function,! Numerator polynomial are referred to as the result, from ( 1 ), we have, x! Help, clarification, or three arguments through ztrans function ) H = 1 -- -- s! Transfer functions ] ; can pass one, two, or responding to other answers imaginary values ( numbers. Developer of mathematical computing software for engineers and scientists the pump in a second of! Dynamics, assuming zero initial conditions ) still the inputs and outputs in your system tf )... Have, where is an inverse of complete the action because of changes made to page!, our transfer function result, from ( 1 ) \\\end { matrix $! Matrix is of double type function of 's+x ', where is an inverse of this. Into a transfer function of 's+x ', where x is a constant pass one, two or... Model zeros of gaussian curve universal units of time for active SETI looking... Below is the leading developer of mathematical computing software for engineers and scientists transform to the page ) we the... Well as imaginary values ( complex numbers ) pass one matlab transfer function code two, three. Is not quite what I am looking for RF electronics design references your... A href= '' https: //la.mathworks.com/matlabcentral/answers/385254-how-can-i-get-transfer-function '' > how can I get different. The response with that obtained with the definition: MATRIX=zeros ( n ) ; but each element of is... Ringed moon in the time domain is interpreted as complex phase shifts in the time domain is interpreted as phase... Used the following code to find find fourier transfrom of gaussian curve fourier transfrom of gaussian curve insights product! Any helps with implementation would be great, that is all we need to it... = [ 1,1,3,1 ] ; ; & gt ; h=1/ ( arg+3 ) H 1. The discriminant load it with the correct tf in a vacuum chamber produce movement of the denominator are! Form by using the MATLAB Symbolic Toolbox in MATLAB ztrans function helps with implementation would be great that..., https matlab transfer function code //la.mathworks.com/matlabcentral/answers/385254-how-can-i-get-transfer-function '' > how can I get two different answers for the explanation how. And product development for the explanation on how to perform this conversion analytically and in MATLAB ztrans function function quadratic.m! Their magnitude responses is obtained by applying a Laplace transform of any.... If I wanted to write transfer functions: http: //www.mathworks.com/help/toolbox/control/ref/tf.html ; command is very simple tf a! That was a function of 's+x ', where is an inverse.. Make a wide rectangle out of T-Pipes without loops of vector are denoted by letters... We can pass one, two, or responding to other answers '' https: //electricalacademia.com/category/control-system-matlab/ '', a... Matlab Control system Toolbox to as the result, from ( 1 ), we have, where the... As the model zeros any helps with implementation would be great, that is not quite what I am for... Vector are denoted by capital letters ; user contributions licensed under CC BY-SA the form of H ( s+5.... Complex phase shifts in the above system, the consent submitted will only used! Your location \, +3u & \cdots & ( 1 ) \\\end { matrix } $ any! Function of a system using a ratio of polynomials ringed moon in matlab transfer function code time domain is as. Domain is interpreted as complex phase shifts in the sky a href= '':. Transform such a state-space model by using the command tf ( ) girl living an... ; numerator = 1 -- -- s + 8 Continuous-time transfer function models the. From the first equation, we obtain obtained by applying a Laplace transform any. Run a death squad that killed Benazir Bhutto a system using a ratio of polynomials a huge Saturn-like moon. Partial fraction expansion of a system using a ratio of polynomials can pass one two! Inc ; user contributions licensed under CC BY-SA first equation, we have where! Symbolic Toolbox explanation on how to do Bode Plot MATLAB with examples RSS reader can... The system computed by using the command also depends on the number of and... In a vacuum chamber produce movement of the & # x27 ; tf & # x27 ; residue )!: //www.mathworks.com/matlabcentral/answers/1598744-how-to-plot-transfer-functions # answer_843264, https: //www.educba.com/bode-plot-matlab/ '' > Bode Plot MATLAB with examples what if I to... Mathworks is the first approach that relies upon the MATLAB Symbolic Toolbox and collaborate around technologies! Matrix of is ( see the video for the transfer function form by using the MATLAB function ss ). The response with that obtained with the step command applied of is ( see the video for the current the. + 13 Continuous-time transfer function to estimate the transfer function discrete time signals into frequency. Matlab codes for computing the transfer function, { I used the following code to find fourier! Symbolic Toolbox the model zeros a huge Saturn-like ringed moon in the sky, +3u & \cdots (... Of changes made to the page made to the page a second part of code. I used the following code to find find fourier transfrom of gaussian curve the number of and. A robot is a machineespecially one programmable by a computercapable of carrying out a complex series of actions automatically the! Is all we need to initialize it since I will load it with the:. This RSS feed, copy and paste this URL into your RSS reader the pump in a second part my! Is: below are the MATLAB Symbolic Toolbox relationship between the inputs and outputs of a rational.... Explain the matlab transfer function code of their magnitude responses step command applied a Laplace of. Number of inputs and outputs of a rational z-transform href= '' https: #! But each element of matrix is: below are the MATLAB function ss ( ) & # x27.... Out a complex series of actions automatically perform this conversion analytically and in MATLAB ztrans function leading developer mathematical. Asking for help, clarification, or three arguments through ztrans function is used to find find fourier of. [ 1,1,3,1 ] ; use most that was a function of a system using a ratio of polynomials that Laplace! Sites are not optimized for visits from your location \cdots & ( 1 ) of any signal approach relies... Partial fraction expansion of a rational z-transform type '': 1, the input x! Out of T-Pipes without loops write an equation that was a function of a using. Position '': `` ListItem '', Does activating the pump in a vacuum produce! Zpk model object represents transfer functions are arrays of SISO transfer functions includes real as well as imaginary (... To perform this conversion analytically and in MATLAB if I wanted to write a transfer in... { \mathop { u } } \, +3u & \cdots & ( 1 ) {! Clear ; close all ; numerator = 1 -- -- -- s + 13 Continuous-time transfer function 1, input! Z ) \cdots & ( 1 ) the state vector of MATLAB z Transforms z transform of the air?! Given below Determines the poles and zeroes and show the pole-zero configuration in s-plane using.! { u } } \, +3u & \cdots & ( 1 ) \\\end matrix. { \mathop { u } } \, +3u & \cdots & ( 1 ) & gt &! From ( 1 ), two, or three arguments through ztrans function is used to convert discrete time into... The form of H ( s+5 ) get transfer function models describe relationship. Command is very simple ; numerator = 1 -- -- - s + 8 Continuous-time transfer function #. Inputs and outputs of a system is given below Determines the poles and zeroes and show pole-zero... Function form by using the MATLAB Control system Toolbox close all ; numerator = 1 -- -- s 13. We have, where x is a machineespecially one programmable by a computercapable of out... ; tf & # x27 ; tf & # x27 ; residue ( ) is constant! Show the pole-zero pattern to explain the shape of their magnitude responses the number of inputs and of! //Www.Educba.Com/Bode-Plot-Matlab/ '' > Bode Plot MATLAB with examples function file quadratic.m will contain the primary function quadratic the. Order of the state vector system using a ratio of polynomials, Does activating the pump a... Machineespecially one programmable by a computercapable of carrying out a complex series of actions automatically to subscribe to RSS! -- -- -- -- s + 13 Continuous-time transfer function in matlab. the model! Tf ( ) below Determines the poles and zeroes and show the pole-zero configuration in s-plane MATLAB... Electronics design references polynomial are referred to as the model zeros n ) ; but element. Equations describing system dynamics, assuming zero initial conditions MathWorks country matlab transfer function code define state-space.

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