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This example shows how to use robust control techniques to design an active suspension system for a quarter car body and wheel assembly model. In this example, you use design techniques to design a controller for a nominal quarter-car model. Then, you use synthesis to design a robust controller that accounts for uncertainty in the model.

Conventional *passive suspensions* use a spring and damper between the car body and wheel assembly. The spring-damper characteristics are adjusted to emphasize one of several conflicting objectives such as passenger comfort, road holding, and suspension deflection. *Active suspensions* use a feedback-controller hydraulic actuator between the chassis and wheel assembly, which allows the designer to better balance these objectives.

This example uses the quarter-car model of the following illustration to design active suspension control laws.

The mass represents the car chassis (body) and the mass represents the wheel assembly. The spring and damper represent the passive spring and shock absorber placed between the car body and the wheel assembly. The spring models the compressibility of the pneumatic tire. The variables , , and are the car body travel, the wheel travel, and the road disturbance, respectively. The force , which is applied between the body and wheel assembly, is controlled by feedback. This force represents the active component of the suspension system.

The following state-space equations describe the quarter-car dynamics.

The state variables in the system are defined as , , , and .

Define the physical parameters of the system. For this example, use the following values.

mb = 300; % kg mw = 60; % kg bs = 1000; % N/m/s ks = 16000 ; % N/m kt = 190000; % N/m

Use these equations and parameter values to construct a state-space model, `qcar`

, of the quarter-car suspension system.

A = [ 0 1 0 0; ... [-ks -bs ks bs]/mb ; ... 0 0 0 1; ... [ks bs -ks-kt -bs]/mw]; B = [0 0; 0 10000/mb ; 0 0; [kt -10000]/mw]; C = [1 0 0 0; 1 0 -1 0; A(2,:)]; D = [0 0; 0 0; B(2,:)]; qcar = ss(A,B,C,D); qcar.StateName = {'body travel xb (m)';'body vel (m/s)';... 'wheel travel xw (m)';'wheel vel (m/s)'}; qcar.InputName = {'r';'fs'}; qcar.OutputName = {'xb';'sd';'ab'};

The model inputs are the road disturbance, , and actuator force, . The model outputs are the car body travel, , suspension deflection , and car body acceleration .

The transfer function from actuator to body travel and acceleration has an imaginary-axis zero. Examine the zero of this transfer function.

tzero(qcar({'xb','ab'},'fs'))

ans = -0.0000 +56.2731i -0.0000 -56.2731i

The natural frequency of this zero, 56.27 rad/s, is called the *tire-hop frequency*.

The transfer function from the actuator to suspension deflection also has an imaginary-axis zero. Examine this zero.

zero(qcar('sd','fs'))

ans = 0.0000 +22.9734i 0.0000 -22.9734i

The natural frequency of this zero, 22.97 rad/s, is called the *rattlespace frequency*.

Plot the frequency response of the quarter-car model from inputs, , to outputs, . Both zeros are visible on the Bode plot.

bodemag(qcar({'ab','sd'},'r'),'b',qcar({'ab','sd'},'fs'),'r',{1 100}); legend('Road disturbance (r)','Actuator force (fs)','Location','southwest') title({'Gain from road dist (r) and actuator force (fs)';... 'to body accel (ab) and suspension travel (sd)'})

Road disturbances influence the motion of the car and suspension:

Small body acceleration influences passenger comfort.

Small suspension travel contributes to good road handling. Further, limits on the actuator displacement constrain the allowable travel.

Because of the imaginary axis zeros, feedback control cannot improve the response from road disturbance ( ) to body acceleration ( ) at the tire-hop frequency. Similarly, feedback control cannot improve the response from to suspension deflection ( ) at the rattlespace frequency. Moreover, there is an inherent trade-off between passenger comfort and suspension deflection. Any reduction of body travel at low frequency results in an increase of suspension deflection. This trade-off arises because of the relationship and the fact that roughly follows at low frequency (less than 5 rad/s).

The hydraulic actuator used for active suspension control is connected between the body mass, , and the wheel assembly mass, . The nominal actuator dynamics can be represented by the first-order transfer function:

The maximum displacement is 0.05 m.

The nominal actuator model approximates the physical actuator dynamics. You can model variations between the actuator model and the physical device as a family of actuator models. You can also use this approach to model variations between the passive quarter-car model and actual vehicle dynamics. The resulting family of models comprises a nominal model with a frequency-dependent amount of uncertainty.

Create an uncertain model that represents this family of models.

ActNom = tf(1,[1/60 1]); Wunc = makeweight(0.40,15,3); unc = ultidyn('unc',[1 1],'SampleStateDimension',5); Act = ActNom*(1 + Wunc*unc); Act.InputName = 'u'; Act.OutputName = 'fs';

At low frequency, below 3 rad/s, the model can vary up to 40% from its nominal value. Around 3 rad/s, the percentage variation starts to increase. The uncertainty crosses 100% at 15 rad/s, and reaches 2000% at approximately 1000 rad/sec. The weighting function, `Wunc`

, reflects this profile and is used to modulate the amount of uncertainty as a function of frequency. The result `Act`

is an uncertain state-space model of the actuator.

Examine the uncertain actuator model by plotting the frequency response of 20 randomly sampled models from `Act`

.

bodeplot(Act,'b',Act.NominalValue,'r+',logspace(-1,3,120)) title('Nominal and 20 random actuator models')

The plus (+) marker denotes the nominal actuator model. The blue solid lines represent the randomly sampled models.

To use synthesis algorithms, you must express your design objectives as a single cost function to be minimized. For the quarter-car model, the main control objectives are formulated in terms of passenger comfort and road handling. These objectives relate to body acceleration, and suspension travel, . Other factors that influence the control design include:

Characteristics of the road disturbance

Quality of the sensor measurements for feedback

Limits on the available control force

Use weights to model external disturbances and quantify the design objectives, as shown in the following diagram.

The feedback controller uses the measurements and of the suspension travel, , and body acceleration, , to compute the control signal, . This control signal drives the hydraulic actuator. There are three external sources of disturbance:

The road disturbance, , which is modeled as a normalized signal shaped by a weighting function .

Sensor noise on both measurements. This noise is modeled as normalized signals and shaped by weighting functions and .

You can reinterpret the control objectives as a disturbance rejection goal. In this interpretation, the goal is to minimize the impact of the disturbances, , , and , on a weighted combination of suspension travel , body acceleration , and control effort . You can consider the norm (peak gain) as the measure of the effect of the disturbances. Then, you can meet the requirements by designing a controller that minimizes the norm from the disturbance inputs, , , and , to the error signals, , , and .

Create the weighting functions that model the design objectives.

Wroad = ss(0.07); Wroad.u = 'd1'; Wroad.y = 'r'; Wact = 8*tf([1 50],[1 500]); Wact.u = 'u'; Wact.y = 'e1'; Wd2 = ss(0.01); Wd2.u = 'd2'; Wd2.y = 'Wd2'; Wd3 = ss(0.5); Wd3.u = 'd3'; Wd3.y = 'Wd3';

The constant weight `Wroad = 0.07`

models broadband road deflections of magnitude 7 cm. `Wact`

is a highpass filter. This filter penalizes high-frequency content in the control signal, and thus limits control bandwidth. `Wd2`

and `Wd3`

model broadband sensor noise of intensity 0.01 and 0.5, respectively. In a more realistic design, `Wd2`

and `Wd3`

would be frequency dependent to model the noise spectrum of the displacement and acceleration sensors. The inputs and outputs of all weighting functions are named to facilitate interconnection. The notation `u`

and `y`

are shorthand for the `InputName`

and `OutputName`

properties, respectively.

Specify target functions for the closed-loop response of the system from the road disturbance, , to the suspension deflection, , and body acceleration, .

HandlingTarget = 0.04 * tf([1/8 1],[1/80 1]); ComfortTarget = 0.4 * tf([1/0.45 1],[1/150 1]); Targets = [HandlingTarget; ComfortTarget];

Because of the actuator uncertainty and imaginary-axis zeros, the targets attenuate disturbances only below 10 rad/s. These targets represent the goals of passenger comfort (small car body acceleration) and adequate road handling (small suspension deflection).

Plot the closed-loop targets and compare to the open-loop response.

bodemag(qcar({'sd','ab'},'r')*Wroad,'b',Targets,'r--',{1,1000}) grid, title('Response to road disturbance') legend('Open-loop','Closed-loop target') %

The corresponding performance weights and are the reciprocals of the comfort and handling targets. To investigate the trade-off between passenger comfort and road handling, construct three sets of weights, . These weights use a blending parameter, , to modulate the trade-off.

beta = reshape([0.01 0.5 0.99],[1 1 3]); Wsd = beta/HandlingTarget; Wsd.u = 'sd'; Wsd.y = 'e3'; Wab = (1-beta)/ComfortTarget; Wab.u = 'ab'; Wab.y = 'e2';

`Wsd`

and `Wab`

are arrays of weighting functions that correspond to three different trade-offs: emphasizing comfort (
= 0.01), balancing comfort and handling (
= 0.5), and emphasizing handling (
= 0.99).

Connect the quarter-car plant model, actuator model, and weighting functions to construct the block diagram of the plant model weighted by the objectives.

sdmeas = sumblk('y1 = sd + Wd2'); abmeas = sumblk('y2 = ab + Wd3'); ICinputs = {'d1';'d2';'d3';'u'}; ICoutputs = {'e1';'e2';'e3';'y1';'y2'}; qcaric = connect(qcar(2:3,:),Act,Wroad,Wact,Wab,Wsd,Wd2,Wd3,... sdmeas,abmeas,ICinputs,ICoutputs);

`qcaric`

is an array of three models, one for each value of the blending parameter,
. Also, the models in `qcaric`

are uncertain, because they contain the uncertain actuator model Act.

Use `hinfsyn`

to compute an
controller for each value of the blending parameter,
. `hinfsyn`

ignores the uncertainty in the plant models and synthesizes a controller for the nominal value of each model.

ncont = 1; nmeas = 2; K = ss(zeros(ncont,nmeas,3)); gamma = zeros(3,1); for i=1:3 [K(:,:,i),~,gamma(i)] = hinfsyn(qcaric(:,:,i),nmeas,ncont); end

The weighted plant model has one control input (`ncont`

), the hydraulic actuator force. The model also has two measurement outputs (`nmeas`

), which give the suspension deflection and body acceleration.

Examine the resulting closed-loop
norms, `gamma`

.

gamma

gamma = 0.9410 0.6724 0.8877

The three controllers achieve closed-loop norms of 0.94 (emphasizing comfort), 0.67 (balancing comfort and handling), and 0.89 (emphasizing handling).

Construct closed-loop models of the quarter-car plant with the synthesized controller, corresponding to each of the three blending parameter values. Compare the frequency response from the road disturbance to , , and for the passive and active suspensions.

K.u = {'sd','ab'}; K.y = 'u'; CL = connect(qcar,Act.Nominal,K,'r',{'xb';'sd';'ab'}); clf bodemag(qcar(:,'r'),'b', CL(:,:,1),'r-.', ... CL(:,:,2),'m-.', CL(:,:,3),'k:',{1,140}) grid legend('Open-loop','Comfort','Balanced','Handling','location','SouthEast') title('Body travel, suspension deflection, and body acceleration due to road')

The solid blue line corresponds to the open-loop response. The other lines are the closed-loop frequency responses for the different comfort and handling blends. All three controllers reduce suspension deflection and body acceleration below the rattlespace frequency (23 rad/s).

To further evaluate the three designs, perform time-domain simulations using the following road disturbance signal :

This signal corresponds to a road bump of height 5 cm.

Create a vector that represents the road disturbance.

t = 0:0.0025:1; roaddist = zeros(size(t)); roaddist(1:101) = 0.025*(1-cos(8*pi*t(1:101)));

Build the closed-loop model using the synthesized controller.

SIMK = connect(qcar,Act.Nominal,K,'r',{'xb';'sd';'ab';'fs'});

`SIMK`

is a model array containing three closed-loop models, one for each of the three blending parameter values. Each model in the array represents a closed loop that is built from the original quarter-car plant model, the nominal actuator model, and the corresponding synthesized controller.

Simulate and plot the time-domain response of the closed-loop models to the road disturbance signal.

p1 = lsim(qcar(:,1),roaddist,t); y1 = lsim(SIMK(1:4,1,1),roaddist,t); y2 = lsim(SIMK(1:4,1,2),roaddist,t); y3 = lsim(SIMK(1:4,1,3),roaddist,t); clf subplot(221) plot(t,p1(:,1),'b',t,y1(:,1),'r.',t,y2(:,1),'m.',t,y3(:,1),'k.',t,roaddist,'g') title('Body travel') ylabel('x_b (m)') subplot(222) plot(t,p1(:,3),'b',t,y1(:,3),'r.',t,y2(:,3),'m.',t,y3(:,3),'k.',t,roaddist,'g') title('Body acceleration') ylabel('a_b (m/s^2)') % configure legend h = legend('Open-loop','Comfort','Balanced','Suspension','Road Dist.',... 'Location','southeast'); h.FontSize = 7; h.Box = 'off'; h.Position = [0.68, 0.58, 0.26, 0.2]; subplot(223) plot(t,p1(:,2),'b',t,y1(:,2),'r.',t,y2(:,2),'m.',t,y3(:,2),'k.',t,roaddist,'g') title('Suspension deflection') xlabel('Time (s)') ylabel('s_d (m)') subplot(224) plot(t,zeros(size(t)),'b',t,y1(:,4),'r.',t,y2(:,4),'m.',t,y3(:,4),'k.',t,roaddist,'g') title('Control force') xlabel('Time (s)') ylabel('f_s (N)')

The simulations show that the body acceleration is smallest for the controller emphasizing passenger comfort. Body acceleration is largest for the controller emphasizing suspension deflection. The balanced design achieves a good tradeoff between body acceleration and suspension deflection.

So far you designed
controllers that meet the performance objectives for the nominal actuator model. However, this model is only an approximation of the true actuator. To make sure that controller performance is maintained even with model error and uncertainty, you must design the model to have *robust performance*. In this part of the example, you use
-synthesis to design a controller that achieves robust performance for the entire family of actuator models that takes uncertainty into account.

Use D-K iteration to synthesize a controller for the quarter-car model with actuator uncertainty.

[Krob,~,RPmuval] = dksyn(qcaric(:,:,2),nmeas,ncont);

The model `qcaric(:,:,2)`

is the weighted quarter-car model for the uncertain model that corresponds to the blending variable
= 0.5.

Examine the resulting -synthesis controller.

size(Krob)

State-space model with 1 outputs, 2 inputs, and 11 states.

Build the closed-loop model using the robust controller, `Krob`

.

Krob.u = {'sd','ab'}; Krob.y = 'u'; SIMKrob = connect(qcar,Act.Nominal,Krob,'r',{'xb';'sd';'ab';'fs'});

Simulate and plot the nominal time-domain response to a road bump with the robust controller.

p1 = lsim(qcar(:,1),roaddist,t); y1 = lsim(SIMKrob(1:4,1),roaddist,t); clf subplot(221) plot(t,p1(:,1),'b',t,y1(:,1),'r',t,roaddist,'g') title('Body travel'), ylabel('x_b (m)') subplot(222) plot(t,p1(:,3),'b',t,y1(:,3),'r') title('Body acceleration'), ylabel('a_b (m/s^2)') subplot(223) plot(t,p1(:,2),'b',t,y1(:,2),'r') title('Suspension deflection'), xlabel('Time (s)'), ylabel('s_d (m)') subplot(224) plot(t,zeros(size(t)),'b',t,y1(:,4),'r') title('Control force'), xlabel('Time (s)'), ylabel('f_s (N)') h = legend('Open-loop','Robust design','Location','southeast'); h.Box = 'off'; h.FontSize = 8;

These responses are similar to those obtained with the balanced controller.

Examine the effect of the robust controller on variability caused by model uncertainty. To do so, simulate the response to a road bump for 120 actuator models randomly sampled from the uncertain model, `Act`

. Perform this simulation for both the
and the robust controllers, to compare the results.

Compute an uncertain closed-loop model with the balanced
controller, `K`

. Sample this model, simulate the sampled models, and plot the results.

CLU = connect(qcar,Act,K(:,:,2),'r',{'xb','sd','ab'}); nsamp = 120; rng('default'); figure(1) clf lsim(usample(CLU,nsamp),'b',CLU.Nominal,'r.',roaddist,t) title('Nominal "balanced" design')

Compute an uncertain closed-loop model with the balanced robust controller, `Krob`

. Sample this model, simulate the sampled models, and plot the results.

CLU = connect(qcar,Act,Krob,'r',{'xb','sd','ab'}); figure(2) clf lsim(usample(CLU,nsamp),'b',CLU.Nominal,'r.',roaddist,t) title('Robust "balanced" design')

The robust controller reduces variability caused by model uncertainty, and delivers more consistent performance.

The robust controller `Krob`

has eleven states. It is often the case that controllers synthesized with `dksyn`

have high order. You can use the model reduction functions to find a lower-order controller that achieves the same level of robust performance. Use `reduce`

to generate approximations of various orders. Then, use `robgain`

to compute the robust performance margin for each reduced-order approximation.

Create an array of reduced-order controllers.

NS = order(Krob); StateOrders = 1:NS; Kred = reduce(Krob,StateOrders);

`Kred`

is a model array containing a reduced-order controller of every order from 1 up to the original 11 states.

Next use `robgain`

to compute the robust performance margin for each reduced-order approximation. The performance goals are met when the closed-loop gain is less than
. The robust performance margin measures how much uncertainty can be sustained without degrading performance (exceeding
). A margin of 1 or more indicates that we can sustain 100% of the specified uncertainty.

% Compute robust performance margin for each reduced controller gamma = 1; CLP = lft(qcaric(:,:,2),Kred); for k=1:NS PM(k) = robgain(CLP(:,:,k),gamma); end % Compare robust performance of reduced- and full-order controllers PMfull = PM(end).LowerBound; plot(StateOrders,[PM.LowerBound],'b-o',... StateOrders,repmat(PMfull,[1 NS]),'r'); title('Robust performance margin as a function of controller order') legend('Reduced order','Full order','location','SouthEast')

The robust performance margin is well below 1 for controllers of order 7 and lower. However there is no significant difference in performance margin between the 8th- and 11th-order controllers, so you can safely replace `Krob`

by its 8th-order approximation.

Krob8 = Kred(:,:,8);

You now have a simplified controller, `Krob8`

, that provides robust control with a balance between passenger comfort and handling.

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