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An Active Disturbance Rejection Control Solutionfor the Two-Mass-Spring Benchmark Problem

Han Zhang1, Shen Zhao2 and Zhiqiang Gao1,3

Abstract— The feasibility of a systematic and effective controldesign for highly uncertain dynamic systems is tested on thewell-known two-mass-spring benchmark problem, based on theactive disturbance rejection control (ADRC) framework. Theproposed solution is obtained in the absence of a detailedmathematical model, in contrast to all previous model-basedsolutions. In addition to meeting the design criteria, the sim-plicity and ease of tuning make the resulting control algorithmappealing to practicing engineers. Furthermore, it is shown inthis paper that even though ADRC doesn’t require detailedmathematical model of the plant, it can take advantage of it ifone is given. By incorporating the model information into theADRC solution, the observer bandwidth and noise sensitivitycan be reduced without performance degradation, thus makingthe solution more practically appealing. The proposed designis also shown to be insensitive to various uncertainties, such asmodel parameters variations, in both simulation and hardwareexperiment.

I. INTRODUCTION

Vibration suppression or isolation is important in a varietyof industry sectors because mechanical resonance physicallyexists in nearly all mechanical systems, such as the auto-mobile suspension systems [1]. Some resonant modes canbe ignored all together if their natural frequencies are farbeyond the bandwidth of the closed loop control system;others must be dealt with deliberately to avoid vibrationand instability. Initially, passive low-pass filter and notchfilter based solutions [2] were used but the quality of thesolutions is limited because they could be quite sensitiveto the uncertainties in the system dynamics. Many differentmethods were proposed to improve the robustness, for whicha two-mass-spring system, which can be seen as equivalentsystem for most typical vibration systems, was formulatedby Wei and Bernstein in 1990 as a benchmark problem [3].

This benchmark problem had been studied by many re-searchers since then and many solutions had been proposedbased on a variety of techniques, such as H∞ control [4], [5],[6], loop shaping [7], multi-objectives differential evolutionalgorithm [8], maximum entropy and optimal projection [9],[10], robust time-optimal control and many other robustcontrol techniques such as the cost averaging technique [11],robust LQR [12], nonlinear matrix inequalities (NMI) [13],etc.

To address different aspects of the vibration problems,some modified versions of the benchmark systems had also

1Center for Advanced Control Technologies, Electrical and ComputerEngineering, Cleveland State University, Cleveland, OH 44115, USA

2National Superconducting Cyclotron Laboratory, Michigan State Uni-versity, East Lansing, MI 48824, USA

3The Corresponding author. E-mail: [email protected]. Tel:1-216-687-3528

been proposed and studied. A combined feedforward andfeedback solution with shaped input to deal with profiletracking problem is presented in 2010 [14]; some variationsof the benchmark problems in the presence of mechanicalfriction were addressed using integral quadratic constraints[15], [16], two-degree-of-freedom controller [17], etc.

What makes the benchmark problem challenging is theuncertainties in the system dynamics and in the externalforces. Typical in the current solutions is the practice oftreating separately the robustness problem caused by theuncertainties in the internal dynamics, and the rejection ofthe external disturbance. The control problem is artificiallysplit accordingly into two separate problems, i.e. the regu-lation problem and the robustness problem. Regulators weredesigned based on the detailed model of the system and thetolerance of parameter uncertainty is treated using robustcontrol design methods.

In this paper, a unified framework is proposed that dealswith the vibration and robustness issues all at once, in theabsence of a detailed plant model. The design turns onthe key concept known as Active Disturbance RejectionControl (ADRC) [18], [19], [20], [21], where the controlleris designed based on the ideal chained integrator model andthe other dynamics and disturbances are lumped together andtreated as the extended state, which is then estimated in realtime and canceled. The purpose of this paper is to examine ifsuch design philosophy could lead to a practical solution forthe two-mass-spring problem that doesn’t require a detailedmathematical model and that the vibration dynamics can betreated as an internal disturbance to be rejected together withthe external ones.

The ADRC solution as a generic controller does notrequire detailed model information, beyond the integral dy-namics. But as it is applied to a particular system, such asthe one studies in this paper, we normally has at least someknowledge of the system dynamics, given in the form ofmathematical model. It is therefore of interests to explorewhat benefits, if any, can be obtained by incorporating suchknowledge into the ADRC solution, perhaps in reducingthe load on the estimation of the total disturbance. In fact,such improvements were observed in both simulation andexperimental studies in terms of the reduced bandwidth andlower noise sensitivity, consistent with the earlier ADRCstudy of a rotational two-inertia problem with differentsystem dynamics [21].

This paper is organized as follows. The dynamics of thetwo-mass-spring system is described in Section II. The pro-posed solution is presented in Section III. Some simulation

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verification of the solution is given in Section IV and thehardware verification results is given in Section V. Finally,some concluding remarks are given in Section VI.

II. DYNAMICS OF TWO-MASS-SPRING SYSTEM

A. System Dynamics

A general schematic of the two-mass-spring system isgiven in Fig. 1, which consists of two masses m1 and m2

that are free to slide over a frictionless horizontal surface.The masses are attached to one another by means of a lighthorizontal spring of spring constant k, thereby creating aresonant mode of a rather low frequency. A damper can beadded between the two masses with a friction coefficient ofc. The control signal u is the force applied to mass m1 andtwo external disturbance forces w1 and w2 are applied tomasses m1 and m2. The positions of masses m1 and m2

are both measured, each one can be used as the output to becontrolled.

The states of the system are conveniently defined as thedisplacements and velocities of the two masses, respectively,with x1 and x3 denote the displacement and velocity ofmass m1, and x2 and x4 as that of the mass m2. Based onNewton’s second law and Hooke’s law, the system dynamicscan be represented in state-space form as

x1

x2

x3

x4

=

0 0 1 0

0 0 0 1

− km1

km1

− cm1

cm1

km2

− km2

cm2

− cm2

x1

x2

x3

x4

+

0

01

m1

0

(u+ w1) +

0

0

01

m2

w2

y =[c1 c2

] [x1

x2

](1)

where c1 = 1, c2 = 0 indicates that the position of mass m1

is the feedback and, conversely, c1 = 0, c2 = 1 indicatesthat of mass m2 is the feedback.

Fig. 1. Two-mass-spring system with uncertain parameters

B. Open-loop Analysis

The poles of the plants vary with different feedbackpoint and friction situations, as shown in Table I. Note thatin all four situations, there are always two poles at theorigin, representing the pure integration from acceleration to

position in each mass and two additional complex conjugatepoles, which are purely imaginary without friction.

TABLE IPOLE LOCATIONS OF THE SYSTEM IN DIFFERENT SCENARIOS

Friction c = 0 c 6= 0Feedback Point m1 m2 m1 m2

Poles

1.414i 1.414i −1 + i −1 + i−1.414i −1.414i −1− i −1− i

0 0 0 00 0 0 0

The Bode plots of the two-mass-spring system shown inFig. 2 prove to be quite indicative of the system dynam-ics and the difficulty of the control problem, in all fourcombinations: with or without friction, with y = x1 ory = x2 . First, at low frequency (below the anti-resonantfrequency) all four models behave like the ideal double-integrator plant with a phase shift of exactly 180 degree.Intuitively, it illustrates that, when moving slowly, the twomasses can be considered as a rigidly connected whole body.As frequency increases, however, i.e. the bodies move faster,the two masses gradually become disjoint, shown particularlyin the variations in the phase plot.

Note that, for the case without the friction, at aroundresonant frequency there is a 180 degree phase shift betweenthe two masses which indicates that the two masses movingagainst each other. In other words, both x1 and x2 areoscillating. Comparing this to the case where there is friction,the phase difference between x1 and x2 is only 90 degrees,a much less sever case of oscillation.

Based on the Bode plot analysis, it appears that the controlproblem is the most challenging when y = x2 and c = 0because the plant can not be approximated using a lowerorder model and the full dynamics must be dealt with bythe controller. In addition, this is the same scenario used inother papers dealing with this benchmark problem. It is forthese reasons this scenario is chosen for the control designchallenge in this paper.

III. THE MAIN RESULT

In this section, an ADRC controller solution is proposedto control the position of the second mass of system withoutfriction. The key point of the proposed solution is to treat thecollection effect of all unknown dynamics of the system andsystem uncertainties together with external disturbances asan equivalent input disturbance, which is then estimated inreal time, with an extended state observer, and then canceledin the control law. Such an approach does not require adetailed mathematical model; more importantly, it can handlewide range of parameter uncertainties. When an establishedmathematical model is already given, it can be incorporatedinto the design. This hybrid method of combining ADRCwith model information leads to a more practical solutionwhere the bandwidth of observer can be reduced for the sameperformance. When an impulse disturbance w2 is introducedon mass m2, the system based on the proposed design can

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−200

0

200M

ag

nitu

de

(d

B)

10−1

100

101

102

−360

−270

−180

−90

0

Ph

ase

(d

eg

)

Bode Plots of Position vs Input Force

Frequency (rad/s)

c1=1,c

2=0 − With friction

c1=1,c

2=0 − Without friction

c1=0,c

2=1 − With friction

c1=0,c

2=1 − Without friction

Fig. 2. Bode Plots of Position vs Input Force

recover within the time constraint in the requirement ii ofdesign #1 of [3].

A. Design Strategy

When the system does not have friction and the distur-bance is coming from w2, the mathematical model could berewritten as

y(4) =− km1 +m2

m1m2y +

k

m1m2w2 +

1

m2w2 + d

+k

m1m2u

(2)

where d refers to model uncertainties and other externaldisturbances in the system. In ADRC framework, we do nothave to use the exact model and parameters of a systemin order to design controller for it. In this applicationparticularly, we could treat the first four terms of the righthand side

f = −km1 +m2

m1m2y +

k

m1m2w2 +

1

m2w2 + d (3)

as the total disturbance to be estimated, as long as we have ahigh enough observer bandwidth. If the observer bandwidthis limited in some applications by sampling rate or noise,some known system dynamics, i. e. the internal disturbance

f1 = −km1 +m2

m1m2y (4)

could be treated as the known dynamics and incorporatedinto the ESO, thus reducing the total disturbance to beestimated to

f2 =k

m1m2w2 +

1

m2w2 (5)

plus d, thereby reducing the ESO load and the associatedbandwidth requirement.

Rewrite the system model as

y(4) = f + bu (6)

where b = km1m2

is the input gain of the system. Then thecontrol law

u =−f + u0

b(7)

reduces (6) to a cascade integral form plant, i. e. the enforcedplant

y(4) ≈ u0 (8)

which can be easily controlled.

B. Disturbance Estimation: the Black Box Approach

Define states for the system (6) as

x =

y

y

y...y

f

(9)

The state space description of the system is

x = Ax+Bu+ Eh

y = Cx(10)

with A =

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

0 0 0 0 0

, B =

0

0

0

b

0

, C =

[1 0 0 0 0

], E =

0

0

0

0

1

and h = f .

for which the ESO in the form of the Luenberger observeris given as

z = A+Bu+ L(y − y)

y = Cz(11)

For the sake of simplicity and easy tuning, the observer gain

L =[5ωo 10ω2

o 10ω3o 5ω4

o ω5o

]is chosen where all eigenvalues of A − LC are placed at−ωo, denoted as the observer bandwidth. Note that furtheroptimizing the placement of these poles could be a topic forfuture research.

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C. Disturbance Estimation: the Gray Box Approach

Since the total disturbance f = f1 + f2 + d, we have

h = f = f1 + f2 + d = −km1 +m2

m1m2

...y + h′ (12)

where h′ = f2+d. Now, an alternative state space descriptioncan be rewritten as

x = A′x+Bu+ Eh′

y = Cx(13)

where A′ =

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

0 0 0 −km1+m2

m1m20

.

The corresponding ESO can be written as

z = A′x+Bu+ L′(y − y)

y = Cz(14)

with the observer gain

L′ =[l1 l2 l3 l4 l5

]designed so that all eigenvalues of A′−L′C would be placedat the observer bandwidth −ωo. The coefficients of L′ arelisted in Table II where a2 = km1+m2

m1m2.

TABLE IICOEFFICIENTS IN L′

l1 5ωo

l2 −a2 + 10ω2o

l3 −5a2ωo + 10ω3o

l4 a22 − 10a2ω2o + 5ω4

o

l5 5a22ωo − 10a2ω30 + ω5

o

D. Control Design for the Cascade Integral Plant

With the appropriately designed and tuned ESO, the plantis reduced to the ideal form of (8), for which the control lawcan easily be designed as

u0 = k1(r − z1)−5∑

n=2

knzn (15)

where r is the set point and the controller gains can be chosenas K =

[ωc 4ω2

c 6ω3c 4ω4

c 1], to place all closed-loop

poles at −ωc, denoted as the closed-loop bandwidth.

IV. SIMULATION VERIFICATION

In this section, the simulation results of the proposedmethods using MATLAB Simulink are presented.

A. System Parameters and Test Profile

The proposed methods are tested using a set of parametersgiven in [3], with m1 = m2 = 1 kg, k = 1 N/m and c = 0N·s/m. In this case the resonant frequency would be 1.414rad/s.

A unit impulse disturbance is introduced on mass m2 att = 1 second as discussed in requirement ii of design #1 of[3] and the position of mass m2 is the output feedback.

B. The Black Box Approach

In order to meet the requirement that the settling time isless than 15 seconds, the closed-loop bandwidth ωc mustbe higher than 0.65 rad/s. The corresponding controllerparameters are selected in tuning as b = 1, ωc = 1, ωo = 15.The control signal u is limited to [−1, 1] in order to makethe simulation realistic. The simulation result is shown inFig. 3.

0 5 10 15 20 25 30−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

Mass P

ositio

n (

m)

Simulation Result for Proposed Solution

x

1

x2

Fig. 3. Simulation result of the black box approach

The system response meets all the specifications of re-quirement ii of design #1 given in [3]. As for the robustness,the proposed design could handle the parameter uncertaintiesand maintain the performance when k varies in the interval[0.3, 40] and m1 and m2 both vary in the internal [0.02, 4],which is several times better compares to existing designmethods. The simulation result is shown in Fig. 4. Thesimulation results confirm the validity of the black boxapproach where no detailed model information is used incontrol design. More importantly, such solution is also shownto be highly robust against the parameter variations.

C. The Gray Box Approach

With the gray box approach, the controller parameterscould be changed to b = 1, ωc = 1 and ωo = 5 where ωo isreduced to one third of the previous value, with similar per-formance but much less noise sensitivity. This demonstratesthe benefits of using the partial model information when it’savailable. The simulation result is shown in Fig. 5.

V. EXPERIMENT VERIFICATION

In addition to the simulation verification, the proposedcontrol solution to the two-mass-spring benchmark problemis also verified in hardware experiment for the load feedbackcase. The experiments are conducted on the rectilinear plant

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YangZJ

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0 200 400 600−2

−1

0

1

2

3

Time (s)

Ma

ss P

ositio

n (

m)

k=40, m1=1, m

2=4

0 200 400 600−1

0

1

2

Time (s)

Ma

ss P

ositio

n (

m)

k=0.3, m1=1, m

2=1

0 200 400 600−50

0

50

100

Time (s)

Ma

ss P

ositio

n (

m)

k=1, m1=1, m

2=0.025

0 200 400 600−0.5

0

0.5

1

1.5

Time (s)

Ma

ss P

ositio

n (

m)

k=1, m1=4, m

2=1

x1

x2

Fig. 4. Simulation results with model uncertainties

0 5 10 15 20 25 30−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

Mass P

ositio

n (

m)

Simulation Result for Model−Aided Solution

x

1

x2

Fig. 5. Simulation result of the gray box approach

Model 210 from Educational Control Products. The proposedcontrol algorithm is discretized with Euler method and im-plemented using the MATLAB Simulink Real-Time toolboxfor a fast verification.

A. Test Setup

The rectilinear plant Model 210 has two mass carriersheld by linear ball slide bearings, with high resolutionoptical encoders mounted at each mass for the purpose ofposition measurement. The left mass is driven by a brush-lessDC (BLDC) servo motor via rack and pinion transmission.Springs can be installed from the base to the left mass, orthe left mass to the right mass, or the right mass to thebase. A damper could also be installed from the right massto the base. Since we only deal with the two-mass-springbenchmark problem in the experiment, one spring is installedbetween the two masses. There are also brass weights the canbe added onto the two mass carriers to change the weight ofthe two masses.

A personal computer (PC) is used as the target computerrunning the MATLAB Simulink Real-Time real-time oper-ating system (RTOS) to execute the control algorithm. TheMATLAB main program is running on another PC (the hostcomputer), connected to the target computer via Ethernet, to

design, compile and download the algorithm. The experimentresults can be plotted on the monitor of the target computerand can be saved and downloaded onto the host computer.A National Instruments PCI-6221 card, with analog signaloutput and quadrature encoder signal input, is installed in thetarget computer to communicate with the rectilinear plant. Aphoto of the experiment platform is given in Fig. 6.

Fig. 6. Photo of the test setup.

B. System Parameters

The mass carriers are 0.50 kg each, the motor and driverack reflected weight is 0.34 kg, and the removable brassweights are 0.51 kg each. Initially, two brass weights willbe added to the left mass carrier and one will be added to theright mass carrier. Therefore, the total equivalent of the leftcarrier will be 0.50+0.34+0.51×2 = 1.86 kg and the totalequivalent of the right carrier will be 0.50+0.51×1 = 1.01kg. The spring constant of the spring between the two masscarriers is 175 N/m.

The torque constant of the drive motor is 0.086 N·m/A,which can be converted to linear force constant of the systemas 2.58 N/A. The motor is driven by a current mode poweramplifier with a gain of 1.5 A/V. Therefore, the total forceconstant of the system is 3.87 N/V. The encoders has 4000lines per revolution, i. e. 16000 counts per revolution. At0.01159 m encoder pinion pitch radius, the resolution of theposition measurement is 54956 counts/m.

C. Test Results

An initial kick is applied to the right mass carrier as theimpulse disturbance. The controller under test is the graybox approach as described in Section III, with the controllerbandwidth, ωc, and the observer bandwidth, ωo, set to 10rad/s and 120 rad/s respectively. The response of the drivemass and load mass are shown in Fig. 7.

Both drive mass and load mass is driven back to theirinitial position in a short period of time with little oscillation.The robustness of the proposed control solution is thenverified by adding one brass mass block to both of the

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−0.01

−0.005

0

0.005

0.01

Time (s)

Ma

ss P

ositio

n (

m)

Hardware Experiment Result for Model−Aided Solution

x1

x2

Fig. 7. Hardware experiment result of the gray box approach

mass carriers and the response hardly changes as shown inFig. 8. Test results show that the proposed solution is veryinsensitive to model uncertainties.

Note that the steady state error in the hardware experimentresults are due to cogging torque in the hardware system andit will die out over time.

0 0.5 1 1.5 2 2.5 3−0.01

−0.005

0

0.005

0.01

0.015

0.02

Time (s)

Ma

ss P

ositio

n (

m)

Hardware Experiment Result for Model−Aided Solution With Disturbance

x1

x2

Fig. 8. Hardware experiment result of the gray box approach with modeluncertainties

VI. CONCLUDING REMARKS

A novel solution to the two-mass-spring benchmark prob-lem is proposed in this paper. By formulating the problem inthe active disturbance rejection control framework, a controlsolution is presented where no detailed mathematical modelof the system is required. It is shown that the proposedmethod works quiet well and has much wider tolerance ofmodel parameter variations while maintaining stability. Avariation of the standard ADRC solution is also proposedwhere the model information is added to the control law,leading to the three-fold reduction in observer bandwidthwithout performance degradation. The disturbance rejectionability and robustness of the proposed solution is verified inboth simulation and hardware experiment.

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