Robust Control Toolbox 

This example shows how to use uncertain objects in Robust Control Toolbox™ to model uncertain systems and to automate robustness calculations using the robustness analysis tools.
On this page… 

Data Structures for Uncertainty Modeling DC Motor Example with Parameter Uncertainty and Unmodeled Dynamics Electrical and Mechanical Equations 
Data Structures for Uncertainty Modeling
Robust Control Toolbox lets you create uncertain elements, such as physical parameters whose values are not known exactly, and combine these elements into uncertain models. You can then easily analyze the impact of uncertainty on the control system performance.
For example, consider a plant model
where gamma can range in the interval [3,5] and tau has average value 0.5 with 30% variability. You can create an uncertain model of P(s) as in this example:
gamma = ureal('gamma',4,'range',[3 5]); tau = ureal('tau',.5,'Percentage',30); P = tf(gamma,[tau 1])
P = Uncertain continuoustime statespace model with 1 outputs, 1 inputs, 1 states. The model uncertainty consists of the following blocks: gamma: Uncertain real, nominal = 4, range = [3,5], 1 occurrences tau: Uncertain real, nominal = 0.5, variability = [30,30]%, 1 occurrences Type "P.NominalValue" to see the nominal value, "get(P)" to see all properties, and "P.Uncertainty" to interact with the uncertain elements.
Suppose we have designed an integral controller C for the nominal plant (gamma=4 and tau=0.5). To find out how variations of gamma and tau affect the plant and the closedloop performance, we form the closedloop system CLP from C and P.
KI = 1/(2*tau.Nominal*gamma.Nominal); C = tf(KI,[1 0]); CLP = feedback(P*C,1)
CLP = Uncertain continuoustime statespace model with 1 outputs, 1 inputs, 2 states. The model uncertainty consists of the following blocks: gamma: Uncertain real, nominal = 4, range = [3,5], 1 occurrences tau: Uncertain real, nominal = 0.5, variability = [30,30]%, 1 occurrences Type "CLP.NominalValue" to see the nominal value, "get(CLP)" to see all properties, and "CLP.Uncertainty" to interact with the uncertain elements.
We can now generate 20 random samples of the uncertain parameters gamma and tau and plot the corresponding step responses of the plant and closedloop models:
subplot(2,1,1); step(usample(P,20)), title('Plant response (20 samples)') subplot(2,1,2); step(usample(CLP,20)), title('Closedloop response (20 samples)')
Figure 1: Step responses of the plant and closedloop models
The bottom plot shows that the closedloop system is reasonably robust despite significant fluctuations in the plant DC gain. This is a desirable, and common characteristic of a properly designed feedback system.
DC Motor Example with Parameter Uncertainty and Unmodeled Dynamics
Now we'll build on the Control System Toolbox™ DC motor example by adding parameter uncertainty and unmodeled dynamics, and investigating the robustness of the servo controller to such uncertainty.
The nominal model of the DC motor is defined by the resistance R, the inductance L, the emf constant Kb, armature constant Km, the linear approximation of viscous friction Kf and the inertial load J. Each of these components varies within a specific range of values. The resistance and inductance constants range within +/ 40% of their nominal values. We use the ureal function to construct these uncertain parameters:
R = ureal('R',2,'Percentage',40); L = ureal('L',0.5,'Percentage',40);
For physical reasons, the values of Kf and Kb are the same, even if they are uncertain. In this example, the nominal value is 0.015 with a range between 0.012 and 0.019.
K = ureal('K',0.015,'Range',[0.012 0.019]); Km = K; Kb = K;
Viscous friction, Kf, has a nominal value of 0.2 with a 50% variation in its value.
Kf = ureal('Kf',0.2,'Percentage',50);
Electrical and Mechanical Equations
The current in the electrical circuit, and the torque applied to the rotor can be expressed in terms of the applied voltage and the angular speed. Create the transfer function H relating these variables, and make AngularSpeed an output of H for later use:
H = [1;0;Km] * tf(1,[L R]) * [1 Kb] + [0 0;0 1;0 Kf]; H.InputName = {'AppliedVoltage';'AngularSpeed'}; H.OutputName = {'Current';'AngularSpeed';'RotorTorque'};
H is an multiinput, multioutput uncertain system as seen from its display.
H
H = Uncertain continuoustime statespace model with 3 outputs, 2 inputs, 1 states. The model uncertainty consists of the following blocks: K: Uncertain real, nominal = 0.015, range = [0.012,0.019], 2 occurrences Kf: Uncertain real, nominal = 0.2, variability = [50,50]%, 1 occurrences L: Uncertain real, nominal = 0.5, variability = [40,40]%, 1 occurrences R: Uncertain real, nominal = 2, variability = [40,40]%, 1 occurrences Type "H.NominalValue" to see the nominal value, "get(H)" to see all properties, and "H.Uncertainty" to interact with the uncertain elements.
The motor typically drives an inertia, whose dynamic characteristics relate the applied torque to the rateofchange of the angular speed. For a rigid body, this is a constant. A more realistic, but uncertain, model might contain unknown damped resonances. Use the ultidyn object to model uncertain linear timeinvariant dynamics. The nominal value of the rigid body inertia is set to 0.02 and we add 15% dynamic uncertainty in multiplicative form:
J = 0.02*(1 + ultidyn('Jlti',[1 1],'Type','GainBounded','Bound',0.15,... 'SampleStateDim',4));
It is a simple matter to relate the AngularSpeed input to the RotorTorque output through the uncertain inertia, J, using the lft command. The AngularSpeed input equals RotorTorque/(J*s), hence "positive" feedback from the 3rd output to the 2nd input of H is used to make the connection. This results in a system with 1 input (AppliedVoltage) and 2 outputs, (Current and AngularSpeed).
Pall = lft(H, tf(1,[1 0])/J);
Select only the AngularSpeed output for the remainder of the control analysis.
P = Pall(2,:)
P = Uncertain continuoustime statespace model with 1 outputs, 1 inputs, 2 states. The model uncertainty consists of the following blocks: Jlti: Uncertain 1x1 LTI, peak gain = 0.15, 1 occurrences K: Uncertain real, nominal = 0.015, range = [0.012,0.019], 2 occurrences Kf: Uncertain real, nominal = 0.2, variability = [50,50]%, 1 occurrences L: Uncertain real, nominal = 0.5, variability = [40,40]%, 1 occurrences R: Uncertain real, nominal = 2, variability = [40,40]%, 1 occurrences Type "P.NominalValue" to see the nominal value, "get(P)" to see all properties, and "P.Uncertainty" to interact with the uncertain elements.
P is a singleinput, singleoutput uncertain model of the DC motor. For analysis purposes, we use the nominal controller synthesized for the DC motor in the "Getting Started with the Control System Toolbox™" manual.
Cont = tf(84*[.233 1],[.0357 1 0]);
First, let's compare the step response of the nominal DC motor with 20 samples of the uncertain model of the DC motor:
clf step(P.NominalValue,'r+',usample(P,20),'b',3) legend('Nominal','Samples')
Figure 2: Openloop step response analysis
Similarly, we can compare the Bode plot of the openloop nominal (red) and sampled (blue) uncertain models of the DC motor.
om = logspace(1,2,80); Pg = ufrd(P,om); bode(usample(Pg,25),'b',Pg.NominalValue,'r+'); legend('Samples','Nominal')
Figure 3: Openloop Bode plot analysis
ClosedLoop Robustness Analysis
In this section, we analyze the stability and performance robustness of the closedloop DC motor system. Our initial analysis of the nominal closedloop system indicates the nominal closedloop system is very robust with 10.5 gain margin and 54.3 deg of phase margin.
margin(P.NominalValue*Cont)
Figure 4: Closedloop robustness analysis
The loopmargin function provides comprehensive stability analysis for multivariable feedback systems. For a control system with N feedback channels, the loopmargin function returns:
The classical gain and phase margins SM for each individual feedback channel (loopatatime margins)
The disk margins DM for each individual feedback channel. The disk margin for the jth feedback channel indicates by how much the transfer function Lj(s) can vary before this particular loop goes unstable.
The multiloop disk margin MM This indicates how much simultaneous, independent gain and phase variations can be tolerated in each feedback channel before the overall closedloop system goes unstable (same as DM for singleloop control systems).
[SM,DM,MM] = loopmargin(P.NominalValue*Cont);
Classical stability margins
SM
SM = GainMargin: 12.4154 GMFrequency: 16.4229 PhaseMargin: 65.7794 PMFrequency: 2.9349 DelayMargin: 0.3912 DMFrequency: 2.9349 Stable: 1
Disk margin
DM
DM = GainMargin: [0.2791 3.5825] PhaseMargin: [58.8074 58.8074] Frequency: 4.9643
Recall that the DC motor plant is uncertain. In addition to the standard gain and phase margins, we can use the wcmargin function to determine the worstcase gain/phase margins for the plantcontroller feedback loop. The wcmargin function calculates the worstcase disk gain and phase margins for each input/output channel. The worstcase analysis shows a possible degradation of the disk gain and phase margins, which were 11 dB and 59 degs respectively, to 1.2dB and 8 degs, respectively, in the presence of the 5 forms of uncertainty modeled in P.
wcmarg = wcmargin(Pg,Cont)
wcmarg = GainMargin: [0.8704 1.1489] PhaseMargin: [7.9252 7.9252] Frequency: 5.1152 WCUnc: [1x1 struct] Sensitivity: [1x1 struct]
Robustness of Disturbance Rejection Characteristics
The sensitivity function is a standard measure of closedloop performance for the feedback system. Let's compute the uncertain sensitivity function S and compare the Bode magnitude plots for the nominal and sampled uncertain sensitivity function.
S = feedback(1,P*Cont); bodemag(usample(S,20),'b',S.Nominal,'r+'); legend('Samples','Nominal')
Figure 5: Magnitude of sensitivity function S.
In the time domain, the sensitivity function indicates how well a step disturbance can be rejected. Let's sample the uncertain sensitivity function and plot its step response to see the variability in disturbance rejection characteristics (nominal appears in red).
step(usample(S,20),'b',S.Nominal,'r+',3); title('Disturbance Rejection') legend('Samples','Nominal')
Figure 6: Rejection of a step disturbance.
We can use the wcgain function to compute the worstcase value of the uncertain sensitivity function gain (peak across frequency). Alternatively, we can use the wcsens function to compute this value: this value.
Sg = ufrd(S,om); [maxgain,worstuncertainty] = wcgain(Sg); maxgain
maxgain = LowerBound: 7.4236 UpperBound: 7.4385 CriticalFrequency: 5.1152
With the usubs function you can substitute the worstcase values of the uncertainty worstuncertainty into the uncertain sensitivity function S. This gives the worstcase sensitivity function Sworst over the entire uncertainty range. Note that the peak gain of Sworst matches the lowerbound computed by wcgain.
Sworst = usubs(S,worstuncertainty); Sgworst = frd(Sworst,Sg.Frequency); norm(Sgworst,inf) maxgain.LowerBound
ans = 7.4236 ans = 7.4236
Now let's compare the step responses of the nominal and worstcase sensitivity:
step(Sworst,'b',S.NominalValue,'r+',6); title('Disturbance Rejection') legend('Worstcase','Nominal')
Figure 7: Nominal and worstcase rejection of a step disturbance
Finally, let's plot Bode magnitude plots of the nominal and worstcase values of the sensitivity function. Observe that the peak value of Sworst occurs at the frequency maxgain.CriticalFrequency:
bodemag(Sg.NominalValue,'r+',Sgworst,'b'); legend('Worstcase','Nominal') hold on semilogx(maxgain.CriticalFrequency,20*log10(maxgain.LowerBound),'g*') hold off
Figure 8: Magnitude of nominal and worstcase sensitivity