MATLAB CONTROL SYSTEM TOOLBOX IN LTI SYSTEM MODEL ANALYSIS

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1 MATLAB CONTROL SYSTEM TOOLBOX IN LTI SYSTEM MODEL ANALYSIS Asist.univ. Luminiţa Giurgiu Abstract The MATLAB environment has important numerical tools. One of them provides a reliable foundation for control system engineering: Control System Toolbox. The Control System Toolbox is a collection of algorithms, expressed mostly in M-files, which implements comon control system design, analysis, and modeling techniques. This paper discusses how to manipulate and analyze linear time-invariant (LTI) systems using extensive tools offered by Control System Toolbox. 1. LTI system model response Lets examine a single-input, single-output (SISO), continuous, linear time invariant (LTI) system defined by its transfer function: Considering the system H( s ) =, with the help of MATLAB we will calculate his s + s + 4 step response. The step response of the system is the output y(t) in case of step function input, u(t)=1(t). The transfer function can be defined in MATLAB by its numerator and denominator in a polynomial form: num=1, den= s + s + 4. The polynomials are defined by their coefficients in a vector form, by descending order of the poles.» num=;» den=[1 4]; The step response of the system displayed directly by the MATLAB step command:» step(num,den); Or it can be written directly:» step(,[1 4]); It is possible to store the values of the step response function in an array.» y=step(num,den) Typing the name of the variable also displays the values.» y The result is a column vector and its elements are the sampled values of the step response function. The sampling time can be calculated from the time interval 0 < t < 6 and the size of the vector.» n=length(y)»t=6/n The calculated sampling time: T=6/109= The help command shows the exact form of the MATLAB commands.» help step

2 This shows, that there are other ways to use the step command:»[y,x,t]=step(num,den); This way not only the y output values, but the x state variables and the t time vector are also stored. The t vector stores the sampling points. It is calculated by MATLAB based on the dynamics of the system (zeros, poles). In many cases it is better to determine the time interval and sampling time directly.»t=0:0.1:10; This sets the time interval to 10 and the sampling time to 0.1. With this time vector the step response can be calculated.» y=step(num,den,t); This output vector now can be displayed with the plot command.» plot(t,y),grid; The plot command uses linear interpolation between the sampling points. If this interpolation is not necessary the» plot(t,y,'.'); command displays only the sampling points. The maximum of the step response is calculated by» ym=max(y) The steady state value of the step response is» ys=dcgain(num,den) and the percentage overshoot of the output is» yovst=(ym-ys)/ys*100. LTI model structures In order to simplify the commands the Control System Toolbox can also use data-structures. The Linear systems are represented by three main models: transfer function, zero-pole-gain and state space models. m m 1 s + b m 1s b s + b1s + b 0 H tf () s = = n n 1 a ns + a n 1s a s + a 1s + a 0 s + 3s + ( s z1 )( s z )...( s z m ) H zpk () s = k = ( s p1 )( s p )...( s p n ) ( s + 1)( s + ) x& = Ax + Bu 3 1 1, A = =, C = [0 1], D = 0 y = Cx + Du, B 0 0 The above defined H(s) transfer function can be defined as a LTI data-structure:» num=» den=[1, 3, ]» H=tf(num,den) or directly» Htf=tf(,[1, 3, ]) Similarly the other models can be defined.» Hzpk=zpk([],[-1, -],)» A=[-3, -1;, 0], B=[1; 0], C=[0, 1], D=0» Hss=ss(A,B,C,D) The models can be converted to the other models.» Hzpk1=zpk(Htf)» Hss1=ss(Htf)» Htf1=tf(Hss) The models consist of parameters. The parameters can be listed by» get(htf)» get(hzpk)» get(hss) The parameters (properties) can be accessed too. The 'v' flag means that the result is in vector format.

3 » [num1,den1]=tfdata(htf,'v') or they can be accessed directly» num=htf.num{1} The {1} have to be used since the transfer function can represent MIMO (multiple inputmultiple output) systems. For SISO (single input-single output) systems it is always {1}. The {1} identifies cell-array type. The cell-array is a matrix with variable size matrices. For example» ca={1, [1,],[1,,3]}» ca{} The other models can be accessed similarly.» Hzpk.k=, Hzpk.z=[], Hzpk.p=[-1, -]» p1=hzpk.p{1}(1)» a1=hss.a Symbolic data input can be used if the s variable is defined» s=tf('s');» H=1/((s+1)*(s+)) We can apply arithmetic operators to LTI models. The defined operators are: +, -, /, \, ', inv, ^. For example the transfer function of the closed loop can be calculated simply:» Hcl=H/(1+H) The series(..) and feedback(..) functions can accomplish the same operation. 3. Time domain investigation: The Control System Toolbox contains several commands that provide the basic tools for time domain investigation. Define the system H(s) = : s + s + 4» H=/(s^+*s+4)» t=0:0.1:10; Step response: y(t) for u(t)=1(t) step input. All the previously examined versions of the step command can be used» step(h); Impulse response: The impulse response is the output of the system in case of u(t)=δ(t) (Dirac function) input.» impulse(h);» yi=impulse(h,t);» plot(t,yi) Initial condition: The system behavior can also be examined if the initial conditions are not zero. This works only for state space models. For the initial command the system has to be transformed into state space representation.» H=ss(H)» x0=[1, -]» [y,t,x]=initial(h,x0);» plot(t,y),grid The state trajectory can also be examined. The first column of the x matrix contains the first state variable and the second column is the second one.» x1=x(:,1); x=x(:,);» plot(x1,x) Arbitrary input: The output can also be calculated for any input signal. Lets calculate the output of the system, if the input is u(t)= *sin(3*t).» u=*sin(3*t);» yl=lsim(h,u,t); Plot both the input(red) and output(blue).

4 » plot(t,u,'r',t,yl,'b'), grid; 4. Frequency domain investigation: The system behavior can also be investigated in frequency domain. Bode diagram of the system can be calculated by the bode command. There are several ways to use this command. The gain and phase shift of the system can be calculated on a fixed frequency point.»w=3 %w means omega» [gain,phase]=bode(h,w) The bode diagram can also be displayed directly. In this case MATLAB calculates a frequency vector automatically based on the system dynamics (similarly to the step command).» bode(h),grid If it is necessary the result of the calculation can be stored in arrays. The frequency response of the system is sampled on a logarithmic scale, since this will give a more viewable form. Nyquist diagram of the system can similarly be generated, except that the transfer function is displayed on the complex plain.» nyquist(h) Margin command is an important tool for examining the stability of the system» margin(h) Zeros, poles: The characteristic polynomial is the denominator of the transfer function, and its roots are the poles of the system.» [num,den]=tfdata(h,'v')» poles=roots(den) The system zeros are the roots of the numerator of the transfer function.» zeros=roots(num) The zeros and poles can be gained from the zpk model» [z,p,k]=zpkdata(h,'v') The zeros and poles can be plotted also on the complex plain by the pzmap command.» subplot(111)» pzmap(h) The damp command lists all the poles and in case of complex pole-pairs the natural frequencies and the damping factors.» damp(h)

5 The DC (zero frequency) gain of the system also be calculated.» dcg=dcgain(h) 5. LTI Viewer A linear system can be investigated in detail by the LTI Viewer. The LTI Viewer is a graphical user interface for examining the system response in time and frequency domain. The systems and the curves can be manipulated from the menus and by the right mouse button.» ltiview or» ltiview('bode',h); 6. Simulink SIMULINK is a graphical software package that makes it possible to simply examine the behaviour of various systems. When creating a new file and copying the different blocks, the parameters of the blocks should be changed to the required values. SIMULINK uses the variables defined in the MATLAB workspace. H(s): Control System Toolbox >LTI system : H Difference: Simulink >Math >Sum: +- Dead time, delay: Simulink >Continuous >Transport Delay: 1 Gain: Simulink >Math >Gain: 1 Step input: Simulink >Sources >Step Scope: Simulink >Sinks >Scope Clock: Simulink >Sources >ClockOutput, time: Simulink >Sinks >To Workspace: y,t The result can be examined directly by the Scope block or it can be send back to the MATLAB workspace by the ToWorkspace output block. The result there can be further processed and display graphically. Changing the Gain parameter between 0.5 and we can determine for example the parameter value for which the system produces constant oscillation. 7. Inverse Laplace Transform Lets calculate the inverse Laplace transform of Y(s) in analytical form. 1 () t k L kl r L 1 re s + p r ( s + p) L 1 pt rte pt 3s ; Y(s) = ( )( ) + 13s + 16 s + s + 3 The function have to be converted to sum of such components whose inverse Laplace transform are known. This partial fraction conversion can be done by the residue command. Define the function in polynomial form.» num=[ ];» den=poly([-3-3 -]); The partial fraction:» [r,p,k]=residue(num,den) r =

6 p = k = [] The partial fraction form in Laplace domain y(t) = e 3t 4te 3t + e t r() 1 r( ) r( 3) 1 4 Y(s) = k = + s p() 1 ( s p( ) ) s p() 3 s + 3 ( s + 3) s + and in time domain. The time function can be calculated from the analytical expression:» t=0:0.05:6;» y=r(1)*exp(p(1)*t)+r()*t.*exp(p()*t)+r(3)*exp(p(3)*t); (The.* means vector element multiplication). This can be also calculated numerically.»yi=impulse(num,den,t); References [1] Norman, S., Control System Engineering, third edition pre-printed at ee.wustl.edu, 000. [] Buckman, Control System Basics, Course Notes at [3] White, J.R., System Dynamics Design and Simulation of Controlled Systems, Online Courses at [4] Glad, T., & Ljung, L., Control Theory, Taylor and Francis, 000. [5] Advanced Control System at