Consider the following transfer function.

sys = tf(3,[1 2 3])

sys = 3 ------------- s^2 + 2 s + 3 Continuous-time transfer function.

To compute the response of this system to an arbitrary input signal, providelsimwith a vector of the timestat which you want to compute the response and a vectorucontaining the corresponding signal values. For instance, plot the system response to a ramping step signal that starts at 0 at timet = 0, ramps from 0 att = 1to 1 att = 2, and then holds steady at 1. Definetand compute the values ofu。

t = 0:0.04:8;% 201 pointsu = max(0,min(t-1,1));

Uselsimwithout an output argument to plot the system response to the signal.

lsim(sys,u,t) gridon

The plot shows the applied input(u,t)in gray and the system response in blue.

Uselsimwith an output argument to obtain the simulated response data.

y = lsim(sys,u,t); size(y)

ans =1×2201 1

The vectorycontains the simulated response at the corresponding times int。

Usegensig(Control System Toolbox)to create periodic input signals such as sine waves and square waves for use withlsim。Simulate the response to a square wave of the following SISO state-space model.

A = [-3 -1.5; 5 0]; B = [1; 0]; C = [0.5 1.5]; D = 0; sys = ss(A,B,C,D);

For this example, create a square wave with a period of 10 s and a duration of 20 s.

[u,t] = gensig("square",10,20);

gensigreturns the vectortof time steps and the vectorucontaining the corresponding values of the input signal. (If you do not specify a sample time fort,ngensiggenerates 64 samples per period.) Use these withlsimand plot the system response.

lsim(sys,u,t) gridon

The plot shows the applied square wave in gray and the system response in blue. Calllsimwith an output argument to obtain the response values at each point int。

[y,~] = lsim(sys,u,t);

When you simulate the response of a discrete-time system, the time vectortmust be of the formTi:dT:Tf, wheredTis the sample time of the model. Simulate the response of the following discrete-time transfer function to a ramp step input.

sys = tf([0.06 0.05],[1 -1.56 0.67],0.05);

This transfer function has a sample time of 0.05 s. Use the same sample time to generate the time vectortand a ramped step signalu。

t = 0:0.05:4; u = max(0,min(t-1,1));

Plot the system response.

lsim(sys,u,t)

To simulate the response of a discrete-time system to a periodic input signal, use the same sample time withgensigto generate the input. For instance, simulate the system response to a sine wave with period of 1 s and a duration of 4 s.

[u,t] = gensig("sine",1,4,0.05);

Plot the system response.

lsim(sys,u,t)

lsimallows you to plot the simulated responses of multiple dynamic systems on the same axis. For instance, compare the closed-loop response of a system with a PI controller and a PID controller. Create a transfer function of the system and tune the controllers.

H = tf(4,[1 10 25]); C1 = pidtune(H,'PI'); C2 = pidtune(H,'PID');

Form the closed-loop systems.

sys1 = feedback(H*C1,1); sys2 = feedback(H*C2,1);

Plot the responses of both systems to a square wave with a period of 4 s.

[u,t] = gensig("square",4,12); lsim(sys1,sys2,u,t) gridonlegend("PI","PID")

By default,lsimchooses distinct colors for each system that you plot. You can specify colors and line styles using theLineSpecinput argument.

lsim(sys1,"r--",sys2,"b",u,t) gridonlegend("PI","PID")

The firstLineSpec"r--"specifies a dashed red line for the response with the PI controller. The secondLineSpec"b”specifies a solid blue line for the response with the PID controller. The legend reflects the specified colors and line styles. For more plot customization options, uselsimplot。

In a MIMO system, at each time stept,inputu(t)is a vector whose length is the number of inputs. To uselsim, you specifyuas a matrix with dimensionsNt-by-Nu, whereNuis the number of system inputs andNtis the length oft。In other words, each column ofuis the input signal applied to the corresponding system input. For instance, to simulate a system with four inputs for 201 time steps, provideuas a matrix of four columns and 201 rows, where each rowu(i,:)is the vector of input values at theith time step; each columnu(:,j)is the signal applied at thejth input.

Similarly, the outputy(t)computed bylsim是一个矩阵的列表示信号在哪里each system output. When you uselsimto plot the simulated response,lsimprovides separate axes for each output, representing the system response in each output channel to the inputu(t)applied at all inputs.

Consider the two-input, three-output state-space model with the following state-space matrices.

A = [-1.5 -0.2 1.0; -0.2 -1.7 0.6; 1.0 0.6 -1.4]; B = [ 1.5 0.6; -1.8 1.0; 0 0 ]; C = [ 0 -0.5 -0.1; 0.35 -0.1 -0.15 0.65 0 0.6]; D = [ 0.5 0; 0.05 0.75 0 0]; sys = ss(A,B,C,D);

Plot the response ofsysto a square wave of period 4 s, applied to the first inputsysand a pulse applied to the second input every 3 s. To do so, create column vectors representing the square wave and the pulsed signal usinggensig。Then stack the columns into an input matrix. To ensure the two signals have the same number of samples, specify the same end time and sample time.

Tf = 10; Ts = 0.1; [uSq,t] = gensig("square",4,Tf,Ts); [uP,~] = gensig("pulse",3,Tf,Ts); u = [uSq uP]; lsim(sys,u,t)

Each axis shows the response of one of the three system outputs to the signalsuapplied at all inputs. Each plot also shows all input signals in gray.

By default,lsimsimulates the model assuming all states are zero at the start of the simulation. When simulating the response of a state-space model, use the optionalx0input argument to specify nonzero initial state values. Consider the following two-state SISO state-space model.

A = [-1.5 -3; 3 -1]; B = [1.3; 0]; C = [1.15 2.3]; D = 0; sys = ss(A,B,C,D);

Suppose that you want to allow the system to evolve from a known set of initial states with no input for 2 s, and then apply a unit step change. Specify the vectorx0of initial state values, and create the input vector.

x0 = [-0.2 0.3]; t = 0:0.05:8; u = zeros(length(t),1); u(t>=2) = 1; lsim(sys,u,t,x0) gridon

The first half of the plot shows the free evolution of the system from the initial state values(-0.2 - 0.3)。Att = 2there is a step change to the input, and the plot shows the system response to this new signal beginning from the state values at that time.

When you uselsimwith output arguments, it returns the simulated response data in an array. For a SISO system, the response data is returned as a column vector of the same length ast。For instance, extract the response of a SISO system to a square wave. Create the square wave usinggensig。

sys = tf([2 5 1],[1 2 3]); [u,t] = gensig("square",4,10,0.05); [y,t] = lsim(sys,u,t); size(y)

ans =1×2201 1

The vectorycontains the simulated response at each time step int。(lsimreturns the time vectortas a convenience.)

For a MIMO system, the response data is returned in an array of dimensionsN-by-Ny-by-Nu, whereNyandNuare the number of outputs and inputs of the dynamic system. For instance, consider the following state-space model, representing a three-state system with two inputs and three outputs.

A = [-1.5 -0.2 1.0; -0.2 -1.7 0.6; 1.0 0.6 -1.4]; B = [ 1.5 0.6; -1.8 1.0; 0 0 ]; C = [ 0 -0.1 -0.2; 0.7 -0.2 -0.3 -0.65 0 -0.6]; D = [ 0.1 0; 0.1 1.5 0 0]; sys = ss(A,B,C,D);

Extract the responses of the three output channels to the square wave applied at both inputs.

uM = [u u]; [y,t] = lsim(sys,uM,t); size(y)

ans =1×2201 3

y(:,j)is a column vector containing response at thejth output to the square wave applied to both inputs. That is,y(i,:)is a vector of three values, the output values at theith time step.

Becausesysis a state-space model, you can extract the time evolution of the state values in response to the input signal.

[y,t,x] = lsim(sys,uM,t); size(x)

ans =1×2201 3

Each row ofxcontains the state values[x1,x2,x3]at the corresponding time int。In other words,x(i,:)is the state vector at theith time step. Plot the state values.

plot(t,x)

The example Plot Response of Multiple Systems to Same Input shows how to plot responses of several individual systems on a single axis. When you have multiple dynamic systems arranged in a model array,lsimplots all their responses at once.

Create a model array. For this example, use a one-dimensional array of second-order transfer functions having different natural frequencies. First, preallocate memory for the model array. The following command creates a 1-by-5 row of zero-gain SISO transfer functions. The first two dimensions represent the model outputs and inputs. The remaining dimensions are the array dimensions. (For more information about model arrays and how to create them, seeModel Arrays(Control System Toolbox)。)

sys = tf(zeros(1,1,1,5));

Populate the array.

w0 = 1.5:1:5.5;% natural frequencieszeta = 0.5;% damping constantfori = 1:length(w0) sys(:,:,1,i) = tf(w0(i)^2,[1 2*zeta*w0(i) w0(i)^2]);end

Plot the responses of all models in the array to a square wave input.

[u,t] = gensig("square",5,15); lsim(sys,u,t)

lsimuses the same line style for the responses of all entries in the array. One way to distinguish among entries is to use theSamplingGridproperty of dynamic system models to associate each entry in the array with the correspondingw0value.

sys.SamplingGrid = struct('frequency',w0);

Now, when you plot the responses in a MATLAB figure window, you can click a trace to see which frequency value it corresponds to.

Load estimation data to estimate a model.

load(fullfile(matlabroot,'toolbox','ident','iddemos','data','dcmotordata')); z = iddata(y,u,0.1,'Name','DC-motor');

zis aniddataobject that stores the one-input two-output estimation data with a sample time of 0.1 s.

Estimate a state-space model of order 4 using estimation dataz。

[sys,x0] = n4sid(z,4);

sysis the estimated model andx0is the estimated initial states.

Simulate the response ofsys使用same input data as the one used for estimation and the initial states returned by the estimation command.

[y,t,x] = lsim(sys,z.InputData,[],x0);

Here,yis the system response,tis the time vector used for simulation, andxis the state trajectory.

Compare the simulated responseyto the measured responsez.OutputDatafor both outputs.

subplot(211), plot(t,z.OutputData(:,1),'k',t,y(:,1),'r') legend('Measured','Simulated') subplot(212), plot(t,z.OutputData(:,2),'k',t,y(:,2),'r') legend('Measured','Simulated')

The choice of sample time can drastically affect simulation results. To illustrate why, consider the following second-order model.

w2 = 62.83^2; sys = tf(w2,[1 2 w2]); tau = 1; Tf = 5; Ts = 0.1; [u,t] = gensig("square",tau,Tf,Ts); lsim(sys,u,t)

figure Ts2 = 0.01; [u2,t2] = gensig("square",tau,Tf,Ts2); lsim(sys,u2,t2)