Real-time Switched Model Predictive Control of a Twin Rotor System

Furqan Tahir, Qadeer Ahmed, and Aamer Iqbal Bhatti

Abstract— In this paper, we present a Switched-DecentralizedModel Predictive Control (S-DMPC) scheme for the real-timeflight control of a Twin Rotor System (TRS). Due to highlynonlinear dynamics and unavailability of full-state information,the linearized TRS model contains modelling (linearization)errors which cause poor state-estimation and lead to controllerperformance degradation. In order to address this problem,we propose an MPC-based control scheme in which, theassociated (linear) observer’s gain is switched according tothe level of linearization error present in the model. Throughexperimental results, we show that the proposed S-DMPCscheme significantly improves TRS tracking in comparison tothe Centralized-MPC as well as the individual (non-switched)Decentralized-MPC schemes.

I. INTRODUCTION

The Twin Rotor System (TRS) is a Multi-Input Multi-Ouput (MIMO) aircraft system which is maneuvered byvertical and horizontal rotors. The laboratory TRS is asimplified version of an actual helicopter in that it consists ofonly two Degrees of Freedom (DOF), namely elevation andazimuth. Furthermore, the rotor blades have a fixed angle ofattack and the TRS is maneuvered by controlling the speedsof these two rotors [1].

TRS presents a significant challenge from a control per-spective due to its fast, highly nonlinear and coupled dy-namics. The motion control is further complicated by thepresence of physical constraints, involvement of stiction andpossible variations in the Centre of Gravity (CoG). Trajectorytracking for the TRS has been investigated through variouscontrol techniques in the literature. Lopez et al. [1] havedesigned a TRS controller using feedback linearization.However, the problem has been simplified by implementingfeedback linearization in elevation dynamics while keep-ing azimuth dynamics at zero. Yu and Liu [2] considera combination of sliding mode control and LQR for theTRS. LQR is first designed to control the elevation andazimuth dynamics and then the sliding mode controller isemployed to improve robustness and disturbance rejection.A few control schemes based on MPC have also beenproposed. In [3], a linear MPC scheme has been applied tothe TRS. However, it ignores modelling errors and mainlyfocuses on real-time implementation issues associated withrunning MPC in a (dedicated) xPC target environment. Anadaptive MPC scheme is proposed in [4] which linearizes

F. Tahir is with the Control and Power Group, Department of Electricaland Electronic Engineering, Imperial College London, SW7 2AZ, UnitedKingdom (e-mail: furqan.tahir07@imperial.ac.uk; ftahir49@hotmail.com).

Q. Ahmed is with the Center for Automotive Research, Ohio StateUniversity, USA (e-mail: qadeer62@ieee.org).

A. I. Bhatti is with the Department of Electronic Engineering, M. A.Jinnah University, Islamabad, Pakistan (e-mail: aamer987@gmail.com).

the prediction model at each sampling instant as well as ateach prediction step. Though catering to modelling error, thisscheme assumes all states as measured and therefore, omitsobserver design.

Many MPC formulations for nonlinear processes makeuse of a linearized model due to the associated computa-tional ease and guarantees on stability etc. Therefore, for a(linear) MPC scheme to provide good control performance,it is important that the linearized model is accurate to areasonable degree. Having said that, as pointed out in [4], andverified through real-time results in Section III, due to thehighly nonlinear nature of the TRS dynamics, it is generallynot possible to obtain a single linearized model whichprovides acceptable results at all operating regions withouttaking account of the modelling errors. Moreover, since theMPC scheme requires full-state information, an importantconsideration in the TRS controller design is the dynamics ofthe observer - required to estimate the unmeasured states - inthe face of such modelling error. Furthermore, for successfulreal-time implementation, an additional constraint is on thecomputational complexity of the proposed (online) MPCscheme given the need for fast TRS control.

In this paper, we address the afore-mentioned prob-lem of designing (full-order) state-observer and MPC con-troller to ensure good tracking performance and rejectionof linearization-error/uncertainty while keeping the overallscheme simple enough to be successfully implemented on theTRS in real-time. We first show that centralized MPC, em-ploying a Kalman filter, exhibits poor tracking performanceparticularly in the elevation dynamics. Subsequently, in orderto provide more design flexibility as well as to reduce compu-tational burden, we propose a Decentralized MPC (DMPC)scheme [5] where the elevation and azimuth dynamics arecontrolled by separate MPC controllers. Furthermore, to min-imize the effect of linearization error in the elevation model,we employ an observer-gain switching approach based onthe trade-off between fast and slow observer dynamics (seee.g. [6] and the references therein). In particular, at operatingregions far away from the linearization point, we employ anobserver having a high gain to reject modelling errors andquickly reconstruct the state. On the other hand, at regionscloser to linearization point, we switch to a low gain observerto reject measurement noise. This is achieved in real-timeby designing Switched MPC controllers [7]. The proposedSwitched-Decentralized MPC (S-DMPC) scheme is shownto significantly improve the TRS tracking performance.

This paper is organized as follows. In Section II, thelinear model is derived for the TRS. Section III formulatesthe MPC problem and presents the real-time TRS tracking

52nd IEEE Conference on Decision and ControlDecember 10-13, 2013. Florence, Italy

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results using centralized MPC. In Section IV, the S-DMPCscheme is designed and all real-time results are discussedand compared. Finally, a conclusion is drawn in Section V.

II. TWIN ROTOR MODEL

We consider the HUMUSOFT CE 150 [8] helicoptersystem. The Helicopter has elevation (α) and azimuth (β)as its two degrees of freedom. The control inputs are thevoltages to the two DC motors.

The free body diagrams, showing all the applied torques,for the elevation dynamics (side-view) and azimuth dynamics(top-view), respectively, are given in Fig. 1. The net torqueequations for elevation and azimuth dynamics can thereforebe written as:

I1α = τ1 + τc + τG − τw − τf (1)

I2β = τ2 − τr − τf2 (2)

where I1 and I2 are the moments of inertia for elevation andazimuth respectively. τ1 and τ2 are the main and side rotortorques. τf and τf2 are the frictional torques in elevationand azimuth respectively. τG and τw are the gyroscopic andgraviational torques. τr is the main motor disturbance torqueand τc is the centrifugal torque.

We consider six states for the TRS model as follows: x1and x4 denote the main and side motor speeds, respectively.x2 and x5 are the elevation and azimuth angles, respectively,and x3 and x6 represent angular speed in elevation andazimuth, respectively. Using equations (1) and (2), withsystem parameters [8] as defined in the Appendix along withthe first order differential equations for the main and sidemotor, yields the following nonlinear model for the TRS (thereader is referred to [8] for further details):

x1 =1

T1(−x1 + u1) (3)

x2 = x3 (4)

x3=1

I1(a1x

21+b1x1−B1x3−Tgsinx2−Kgyu1x6cosx2) (5)

x4 =1

T2(−x4 + u2) (6)

x5 = x6 (7)

x6 =1

I2(a2x

24 + b2x4 −B2x6 −KrToru1) (8)

Here, the control input u1 is the main-motor voltage (forelevation) and u2 is the side-motor voltage (for azimuth).Note that in (5) and (8), among other terms, the nonlinearityexists in the terms involving motor speeds x1 and x4. This,as we will show in the following sections, leads to modelling(linearization) error since both x1 and x4 are unmeasured inthe TRS (only the outputs x2 and x5 are measured).

Linearizing equations (3)-(8) around the point x = 0 yieldsthe following continuous-time TRS model:

x = Acx+Bcu, y = Ccx+Dcu (9)

Fig. 1. Free body diagrams for Elevation (side-view) and Azimuth (top-view).

[Ac Bc

Cc Dc

]:=

− 1T1

0 0 0 0 0 1T1

0

0 0 1 0 0 0 0 0b1I1−Tg

I1−B1

I10 0 0 0 0

0 0 0 − 1T2

0 0 0 1T2

0 0 0 0 0 1 0 0

0 0 0 b2I2

0 −B2

I2−KrTor 0

0 1 0 0 0 0 0 00 0 0 0 1 0 0 0

With the parameter values as specified in the Ap-pendix, the eigenvalues of the TRS are given byλc = {−0.21± 2.95i,−3.33, 0,−2.1,−4} and therefore,the system is marginally stable. Furthermore, the realisationof the system is controllable as well as observable - whichmakes the observer design possible.

Note that we can discretize the continuous-time system(9), using e.g. Zero-Order Hold (ZOH), to yield an equivalentdiscrete-time system of the form:

xk+1 = Axk +Buk, yk = Cxk (10)

III. CENTRALIZED MPC SCHEME

In this Section we first formulate the MPC problem andshow the dependence of the MPC control law on full-stateinformation. We then discuss the observer theory. After-wards, the real-time centralized MPC results for the TRSare presented.

A. MPC Formulation

MPC is an optimal control scheme in which the currentcontrol action is obtained by solving on-line, at each sam-pling instant, a finite horizon open-loop optimal control prob-lem. The optimization yields an optimal control sequence andthe first control in this sequence is applied to the plant [9].

Given the system in (10), the MPC algorithm computes,at each sampling instant, the control action which minimizesa cost function of the form:

J :=

Ny∑k=1

(yk − rk)TQk(yk − rk) +Nu−1∑k=0

uTkRkuk (11)

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subject to the constraints:

yk≤ yk ≤ yk, uk ≤ uk ≤ uk (12)

where rk is the desired output, Qk ≥ 0 and Rk > 0 are the(diagonal) weighting matrices penalizing tracking-error andthe control-move respectively. Moreover, Ny and Nu are theprediction and control horizons respectively, with Ny > Nu.

By iterating the dynamics (10) and manipulating the result,the MPC problem can be recast into the following QuadraticProgram (QP) (see e.g. [9] for details):

minCU ≤ d

J(x0, U) (13)

where UT =[uT0 uT1 · · · uTNu−1

]. The QP in (13) is

solved at each time step and the first control in the optimalsequence U? is applied to the plant. Note that the MPCproblem explicitly requires the knowledge of the currentstate (x0). However, in the considered TRS, only the outputs(x2, x5) are measured. Therefore, the observer dynamics areof great significance in MPC and we discuss these next.

B. Observer Theory

For the system in (10), we consider a (full-order) linearstate-observer described by the equations [9]:

yk|k−1 = Cxk|k−1 (14)xk|k = xk|k−1 +M(yk − yk|k−1) (15)

xk+1|k = Axk|k +Buk (16)

Here M represents the observer gain and xk|k denotes theestimated (current) state at time k and therefore replacesx0 in the MPC problem (13). Note that M represents atrade-off between the conflicting requirements of measure-ment noise immunity vs modelling error rejection and faststate reconstruction [6]. Observers with a ‘low-gain’ providegood measurement noise immunity as the ‘predicted state’component dominates the state-estimate in (15). On the otherhand, observers having a ‘high-gain’ produce faster statereconstruction by giving a relatively higher weightage tothe ‘correction term’ in (15). This allows them to rejectthe uncertainty/error present in the (prediction) model -though at a cost of reduced (output) measurement noiseimmunity. Using (14)-(16), the dynamics of the estimationerror ek(:= xk|k − xk) can be shown to be given by:

ek+1 = (A− LC)ek (17)

where L = AM . The observer gain M can be designedusing, for example, concepts from Kalman filtering which

TABLE ITRACKING PROFILE

Reference Time Interval (sec)Angle 0− 40 40− 80 80− 120 120− 160

Elevation +14◦ +20◦ +14◦ +7◦

Azimuth −40◦ −50◦ −40◦ −30◦

considers stochastic disturbance/noise in the process and/ormeasurements. Alternatively, deterministic methods such aspole-placement algorithms can also be used to choose M sothat poles of error dynamics (17) are at desirable locations.

C. Real-time Implementation of Centralized-MPC

For the implementation of Centralized-MPC (C-MPC),we discretize the TRS model (9) - using a sampling timeTs = 0.01 - to obtain a proper, discrete-time model (10). Forthe cost, we found horizons Ny=22 and Nu=5 to be suitablefor implementation. The weighting matrices considered forimplementation are: Qk = qI,Rk = rI ∀k where I is theidentity matrix and the ratio q/r = 60, 000 (chosen to em-phasize good tracking performance). Furthermore, the stateobserver is designed to be the Kalman filter with white noiseconsidered in the measured outputs. Finally, we considerthe following system constraints on the (normalized) inputvoltages and output angles for MPC:

−1 ≤ u1, u2 ≤ 1; 0◦ ≤ α ≤ 30◦; − 80◦ ≤ β ≤ 0◦. (18)

In order to critically evaluate the tracking performanceof MPC, we consider large, simultaneous step-changes inelevation and azimuth references after every 40s. The step-tracking profile considered throughout is given in Table I.

The MATLAB simulation results for C-MPC were idealand are therefore not provided. Fig. 2 shows the results ofreal-time application of C-MPC on the TRS.

We see in Fig. 2 that C-MPC is unable to properly controlthe TRS, particularly in elevation. Analyzing the results,we note that at low elevation setpoint, e.g. 7◦ in the timeinterval [120−160]s, the controller performance is very goodand the elevation angle tracks the reference well. However,the same C-MPC controller struggles to stabilize the TRSat higher setpoints. In particular, the oscillation size growsproportionally with elevation setpoint so much so that, at thehighest setpoint (20◦), the TRS hits both the upper (30◦) andlower (0◦) limits of the setup. Now analyzing the azimuthresults, the azimuth tracking performance is reasonable in thefirst 40s and the last 40s time interval with output settlingdown to within 5% of the reference value. Note that theazimuth oscillations in the time intervals [40 − 80]s and[80 − 120]s can be attributed to the elevation oscillationssince the TRS has coupled dynamics. It is worth mentioninghere that increasing q/r ratio, or changing the horizons etc,did not result in any noticeable improvements in tracking.

IV. SWITCHED-DECENTRALIZED MPC

The major reason behind the poor elevation trackingperformance of C-MPC is the modelling/linearization error.Note that, due to motor speeds being unmeasurable, theTRS nonlinear model (3)-(8) was linearized at the pointx1 = 0 (starting operating condition). Hence, the modelis reasonably accurate only at lower motor speeds - andtherefore, at lower elevation angles. This explains the goodelevation tracking performance in the time interval [120 −160]s of Fig. 2. However, at higher elevation angles (andtherefore, higher motor speeds), the linear model (9) becomes

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Fig. 2. Real-time results with Centralized MPC

inaccurate and involves larger modelling error which leads topoor state-estimation and, in turn, deterioration of trackingperformance. This phenomenon can be observed in Fig.2. In order to remedy the afore-mentioned problem, wepropose a Decentralized MPC (DMPC) framework [5] inwhich elevation and azimuth dynamics are controlled bytwo separate MPC controllers. This provides more designflexibility, e.g. observer design is easier for SISO systems.

We first decompose the MIMO system (9) into two SISOsystems. Therefore, for elevation and azimuth, we obtain thefollowing subsystems dynamics, respectively:

xe =

x1x2x3

=− 1

T10 0

0 0 1b1I1

−Tg

I1cos (αo) −B1

I1

x1x2x3

+ 1

T1

00

u1(19)

xa =

x4x5x6

=

− 1T2

0 0

0 0 1b2I2

0 −B2

I2

x4x5x6

+

1T2

00

u2 (20)

where αo is the operating elevation angle in radians (previ-ously assumed 0 in the linearization for (9)).

Note that, in the interest of algorithmic simplicity, we haveignored the interaction (coupling) between the elevation andazimuth dynamics. Of course, more sophisticated schemessuch as Distributed MPC - which incorporates communica-tion amongst controllers to improve overall performance androbustness - can be designed to take account of coupling.However, the associated computational burden may prohibitits successful real-time application to a fast dynamical systemsuch as the TRS.

A. Elevation ControllerFor the control of elevation dynamics (19), we design a

Switched-Decentralized MPC (S-DMPC) scheme by dividingthe elevation operating range (α) into three Control Regions(CRs), namely:

CR1 : 0◦ ≤ α ≤ 9◦, αo = 4.5◦

CR2 : 9◦ < α ≤ 18◦, αo = 13.5◦

CR3 : 18◦ < α ≤ 30◦, αo = 22◦

For CRs 1 and 2, we obtain linear models by substitutingin (19), the mean elevation angles (after conversion intoradians) i.e. αo = 4.5◦ and αo = 13.5◦ respectively. Whereasfor CR3, we substitute αo = 22◦. Furthermore, we designthree (elevation) controllers, based on these models, such thateach DMPCi corresponds to CRi (for i = 1, 2, 3).

As mentioned earlier, the main source of modelling (lin-earization) error is motor speed (x1). However, since x1 isunmeasureable in the considered TRS setup, we are unableto linearize the prediction/estimation model at various x1operating points. This causes convergence problems for theobserver and, in turn, for the whole MPC scheme, whichis heavily reliant upon the estimated state. We address thisissue by designing suitable observer with a Low/High-gainfor the controller (DMPCi) of each CRi as follows. Here,note that the separation principle holds as both the observerand control scheme are linear.

First of all, to improve estimation, we consider an in-tegrated white noise in the measured output - sometimesalso called constant output disturbance [9]. The observerdynamics for elevation are therefore modified by augmentingthe estimation model in (14) and (16) as follows:

Ad =

[A 00 1

], Bd =

[B0

], Cd =

[C 1

]For CR1, the TRS operation is closer to the linearizationpoint (due to small x1) and hence involves relatively smallmodelling error. Therefore, focusing on noise rejection prop-erties, we design a low-gain observer by placing the poles ofthe error dynamics (Ad−LCd) nearer the unit disk. In par-ticular, for DMPC1, we design an observer with eigenvaluesλ1 = {0.860, 0.965± 0.024i, 0.998} yielding the observergain L1 = [0.173, 0.154, 0.442, 0.019]

T . Note that eigen-values λ1 have been chosen to be (slightly) smaller thanthose achieved with a Kalman filter. This is to reduce the(large) undershoot which occurs at time t = 122s in Fig. 2.

Now considering DMPC3, we note that the modellingerror will be large owing to the high motor speed x1required to elevate the TRS in the CR3 range. Therefore,in order to reject this modelling error and to quickly recon-struct the state, we design an observer with a high gain,by placing the poles of the error dynamics closer to theorigin. In particular, we substantially reduce the eigenvalueassociated with the estimation error in x1 by choosingλ3 = {0.06, 0.94± 0.01i, 0.998}. The corresponding ob-server gain is given as L3 = [2.065, 0.737, 7.925, 0.288]

T .Finally, for DMPC2, we place the observer poles

in between λ1 and λ3 by choosing eigenvaluesλ2 = {0.40, 0.962± 0.01i, 0.998}. This yields theobserver gain L2 = [0.195, 0.567, 2.365, 0.073]

T . Notethat, we have kept the eigenvalue for ‘integrated whitenoise’ quite large (at 0.998) in all three observers.

Using the elevation angle measurement, a signal is gener-ated to switch between the three DMPC controllers so as tooperate them in their respective CRs. To avoid the peakingphenomenon as well as to ensure bumpless transfer duringsuch switching, the real-time input/output measurements are

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Fig. 3. Real-time results with S-DMPC, and (non-switched) DMPC1, DMPC2 and DMPC3 respectively

simultaneously provided to all three DMPC controllers. Thisallows all controllers (both active and inactive) to continu-ously update their state estimate - enabling smooth transition.

B. Azimuth Controller

The azimuth dynamics (20) are only marginally stablewhich prohibits the inclusion of the constant output dis-turbance in the observer dynamics (as we considered forelevation controller observers). Therefore, in order to providea reasonable state-reconstruction speed, we design a (non-switched) DMPC controller with observer eigenvalues placedat λa = {0.977, 0.677± 0.64i}. This results in the observergain La = [−6.01, 0.10, 52]

T .

C. Real-time Implementation of Switched DMPC

For comparison, we set the tuning parameters Ny , Nu,Qk and Rk, for both elevation and azimuth controllers to beexactly the same as those used for C-MPC. Furthermore, we

test the algorithm using the same tracking profile (Table I).The real-time elevation tracking results for S-DMPC alongwith those obtained when DMPC1, DMPC2 and DMPC3are used throughout individually (without switching) areshown in Fig. 3. A quantitative analysis of Fig. 3, includingOvershoots (OS) and Undershoots (US) in degrees as wellas the 5% settling time (Ts), is provided in Table II.

From Fig. 3 and Table II, it is clear that S-DMPC deliversthe best elevation tracking performance in terms of overshootas well as the settling time. Furthermore, the region ofoperation remains well within the TRS constraints (18).Note that with DMPC1 controller used throughout, the low-gain observer, though exhibiting good measurement noiseimmunity, provides slow state reconstruction and fails toconsider the modelling error - especially at high elevationangles. This, in turn, causes large overshoots - most notablyat around t = 41s when the TRS nearly hits the system’s

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TABLE IIQUANTITATIVE ANALYSIS OF REAL-TIME ELEVATION RESULTS

Time Interval (sec)

0− 40 40− 80 80− 120 120− 160

Controller OS Ts OS Ts US Ts US Ts

S-DMPC 3.12 5.32 5.18 1.05 4.26 8.6 3.08 10.8

DMPC1 5.36 7.03 9.88 3.2 6.67 8.66 4.15 10.3

DMPC2 6.37 9.49 7.58 3.66 4.15 8.74 - -

DMPC3 - - 5.23 1.04 - - - -

C-MPC - - - - - - 7 11.5

upper limit. On the other hand, when DMPC3 controller isused throughout, its observer (with higher gain) achieves faststate convergence and good modelling error rejection with acompromise on measurement noise immunity. Consequently,we see good performance at high elevation angles (withlarge modelling error). However, at low elevation angles(with low modelling error), we see oscillations which, amongother factors, are also contributed by measurement noise.Finally, the results with non-switched DMPC2 are a trade-off between those obtained with DMPC1 and DMPC3.

Fig. 4 shows that proposed scheme provides good trackingfor the azimuth dynamics as well. From Fig. 5, we seethat the elevation control effort also remains within the con-straints. It is worth mentioning here that we have deliberatelytested the controllers using the most disturbing (and slightlyunrealistic) tracking profile by considering large, simulta-neous step-changes in elevation and azimuth references.This, expectedly, results in large (though still constraint-admissible) control spikes at each reference change.

In addition to the linearization errors, another source ofmodel-uncertainty is the variation in the Centre of Gravity(CoG) which, in a real helicopter, may be caused by on-boardweight variations or turbulence in the fuel/coolant tanks etc.Fig. 6 shows the real-time results when the TRS CoG isshifted from 0 to 50% for the interval t = 20−45s and thenback to 0. We see that S-DMPC controller is able to handlelarge CoG variations with only minor output deviations fromthe reference (3.1◦ at t = 21s and 3.46◦ at t = 46s).

V. CONCLUSIONThe Twin Rotor System is a MIMO system which suffers

from a high degree of nonlinearity as well as modelling (lin-earization) errors. In order to address the control challengesassociated with this system, we have designed a Switched-Decentralized MPC scheme which takes account of mod-elling errors - at different operating points - through observer-gain switching based on the concepts from high-gain/low-gain observer theory. Moreover, the scheme has low onlinecomputational complexity to be successfully implemented,in real-time, on a fast dynamical system such as the TRS.

Through real-time results, we have shown that the pro-posed S-DMPC scheme substantially improves the trackingperformance as compared to the Centralized MPC as well asthe (non-switched) Decentralized MPC schemes.

Fig. 4. Real-time Azimuth results for S-DMPC

Fig. 5. Control input u1 for S-DMPC

Fig. 6. Real-time CoG variation results with S-DMPC

APPENDIX

TRS Outputs 30 degrees in Elevation80 degrees in Azimuth

TRS Parameters T1 = 0.3 s, T2 = 0.25 s, Tor = 2.7 sa1 = 0.105 N.m/MU, b1 = 0.00936 N.m/MU2

I1 = 4.37e−3 Kg.m2, B1 = 1.84e−3 Kg.m2/sTg = 3.83e−2 N.m, Kgy = 0.015 Kg.m/s

a2 = 0.033 N.m/MU, b2 = 0.0294 N.m/MU2

Kr = 0.00162 N.m/MU, I2 = 4.14e−3 Kg.m2

B2 = 8.69e−3 Kg.m2/s

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