H2 Vs H∞ control of TRMS via output error optimization augmenting sensor and control singularities

Ain Shams Engineering Journal xxx (xxxx) xxx

Contents lists available at ScienceDirect

Ain Shams Engineering Journal journal homepage: www.sciencedirect.com

Electrical Engineering

H2 Vs H1 control of TRMS via output error optimization augmenting sensor and control singularities Parthish Kumar Paul ⇑, Jeevamma Jacob Department of Electrical Engineering, National Institute of Technology Calicut, India

a r t i c l e

i n f o

Article history: Received 30 May 2016 Accepted 11 July 2019 Available online xxxx Keywords: Modeling Singularity Output error optimization Well-posedness TRMS

a b s t r a c t This paper addresses the robust control problem of Twin Rotor Multi input multi output System (TRMS) via H2 and H1 control techniques. An output error optimization technique is proposed to develop H2 and H1 controllers for the well posed plant. Computer simulation results are presented for closed loop TRMS in hovering positions which show marked improvements over previous works. The simulated plant exhibits stable responses in hovering position at the desired pitch and yaw angles. Corrections are incorporated in model formulation to compensate control and sensor singularities. The output error optimization technique proposed in the present paper can be essentially adopted in controlling of 2 by 2 plants exhibiting non-minimum phase dynamics. Ó 2019 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/ by-nc-nd/4.0/).

1. Introduction With the advent of robust control techniques and state-of-theart mathematical tools, the stabilization and control problems of airborne vehicles are less tedious; yet the challenges in designing and proposing control algorithms for such plants are in no way lesser. A class of such Multi Input Multi Output (MIMO) systems with Right Half Plane (RHP) dynamics is difficult to be tackled due to inverse responses. The TRMS, developed by Feedback Instruments Ltd. [1], a laboratory model of helicopter known for its intricate cross coupled and non-minimum phase dynamics is a test bench which can be used to validate robust control techniques. Multiple zeros at the origin bring TRMS to the verge of instability and introduce a wide arena of challenges in robust control applications on the system. The present paper is the result of motivation to propose another robust control technique to fly TRMS. TRMS is a laboratory scale electro mechanical helicopter derived from traditional rotorcraft. It has improved rotor dynamics ⇑ Corresponding author at: Department of Electrical and Electronics Engineering, Dr. Akhilesh Das Gupta Institute of Technology and Management, New Delhi – 110053, India. E-mail addresses: [email protected] (P.K. Paul), [email protected] (J. Jacob). Peer review under responsibility of Ain Shams University.

Production and hosting by Elsevier

resulting in more intricate system mathematics. In human maneuvered rotorcrafts, the control is generally achieved by tilting the blades of the rotors at appropriate angles with the collective and cyclic actuators, while keeping the rotor speed constant as cited in their work by Martinez et al. [2]. The blades of the rotors in TRMS have a fixed angle of attack, and control is achieved by controlling the speeds of the rotors as presented by Martinez et al. [3]. Accordingly, this system presents higher coupling between dynamics of the rigid body and that of the rotors as compared to a conventional helicopter, and yields highly non-linear, strongly cross-coupled dynamics. TRMS has one main rotor in the horizontal plane and one tail rotor in the vertical plane with fewer degrees of freedom of movement and more involved plant dynamics as compared to a conventional helicopter. Thus, this setup may be tested in laboratory for its controllability and stability in two degrees of freedom of movement through two degrees of freedom of control. Hence, this system is fit to do laboratory experiments and understand the effect of cross coupling in 2  2 systems in specific and MIMO systems in general. Fig. 1 illustrates a simple schematic diagram of TRMS at rest. Fig. 2 is illustration of real TRMS in hovering position. Robust control techniques can be implemented to achieve desired pitch and yaw displacements of the TRMS from the rest and acquire hovering positions of the system as well. A robust controller guarantees robust behavior from external disturbances and modeling uncertainties, i.e., controls the slow dynamics robustly, while neglecting the fast dynamics of a system. In TRMS, the propeller-rotor groups exhibit fast dynamics, while movement on

https://doi.org/10.1016/j.asej.2019.07.001 2090-4479/Ó 2019 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article as: P. K. Paul and J. Jacob, H2 Vs H1 control of TRMS via output error optimization augmenting sensor and control singularities2 Vs H1 control of TRMS via output error –>, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.07.001

2

P.K. Paul, J. Jacob / Ain Shams Engineering Journal xxx (xxxx) xxx

Fig. 1. Sketch of TRMS laboratory unit [1].

Fig. 2. Working model of TRMS in hover. (Curtsey: Control Systems Laboratory, Department of Electrical Engineering, National Institute of Technology Calicut, India.)

the pitch and yaw angles are confined to slow dynamics. In addition, the linear models obtained by traditional linearization methods are nominal models and devoid of critical system information. It is also known that a robust controller is able to stabilize the system when there is marked variation between the actual plant and the linear model used to determine the robust controller. Thus, TRMS is a suitable platform for testing robust control strategies. The techniques and algorithms for the design of robust controller for TRMS have been improved gradually as reported in various national and international conferences and journals. In the present work, this class of systems have been extensively surveyed, identifying a number of control strategies that range from linear robust control techniques as stated by Smerlas et al. in [4] to more recent nonlinear approaches as mentioned by Vilchis et al. in [5]. It can be proved that the dynamics of TRMS are non-minimum phase, exhibiting unstable zero dynamics, which make the system not suitable for classical feedback linearization techniques [2]. This fact, jointly with important modeling uncertainties, especially in the high frequency range, makes the system hard to control by standard techniques as mentioned by Mullhaupt et al. in [6]. Other classical approaches to sort out the difficulty associated with modeling uncertainties in the high frequency range, make use of simplifying assumptions to decouple rigid body and rotor dynamics, such that partial feedback linearization can be applied

[3]. With these assumptions, the linearized system takes the form of a classical master-slave cascade linear system, where the simplified linear system can be controlled with well-known linear strategies for instance, H1, LQR, optimal, or switched control [2,6]. Contemporary to these results, several advances in the theory of nonlinear control have been reported. One such control strategy is the nonlinear extension of classical robust L2 control [3]. Lu and Wen [7] have decoupled TRMS as two SISO plants and it is controlled through one degree of freedom controller on each of the SISO system. Several robust control optimization techniques are discussed by Skogestad and Postlethwaite in [8] from which the present work is partially motivated. In this work, an output error optimization technique is proposed by using generalized plant encapsulating TRMS and other subsystems except robust controller. The H2 and H1 optimization algorithm and the generalized plant are used to design the robust controllers. Prior to this, the nonlinear dynamic mathematical model of TRMS is linearized to obtain the nominal model. The generalized plant is formulated by using the nominal model of TRMS. Determination of the nominal model of a physical plant is the first and inevitable step in solving any control system problem in general. In this case, the required nominal model is linear. This can be determined by linearizing the mathematical relationship of the system and obtaining the state space model. Several efforts have been made in the past to derive a representative linear model of TRMS that behaves in the same manner as the physical TRMS in hover (translatory motion w.r.t. ground being zero). In a recent work, Nejjari et al. [9] have presented system identification and modeling of TRMS leading to Quasi-LPV modeling. In a former work, the forward path Transfer Functions (T.F.s) were derived by Lu and Wen [7] for the SISO sub-systems of the decoupled TRMS. A dynamic model characterizing the TRMS in hovering position was extracted using a black box system identification technique in the works of Ahmed et al. [10–13]. Identification of system parameters of TRMS based on least squares parametric identification method is presented by Belkheiri et al. [14]. Tanaka et al. [15] considered linear parameter varying identification of TRMS via grey box modeling. Rahideha et al. [16] and Ekbote et al. [17] had used Lagrangian based formulation of equation of motion to derive a dynamic model of TRMS. In a recent attempt, Vishnupriyan et al. [18] have presented uncertainty modeling and control using linearized plant model of TRMS. In another recent work, a model based on first-principle modeling was derived and its parameters were refined by identification based on real-time experiments as presented by Chapula et al. [19]. In the present work a nominal model of TRMS is obtained by linearizing it using Taylor series expansion followed by defining Jacobians. A 2  2 T.F. model is obtained from the state space representation. But this T.F. model is devoid of the RHP zero dynamics of the plant. The missing dynamics in the T.F. matrix are taken care by robust control technique developed in this work. Subsequently, a generalized output error optimization technique is proposed in this paper. Two different robust controllers are designed using this technique, viz. H2 and H1 controllers. A feed forward gravity compensation block is used to compensate acceleration due to gravity in the vertical plane. The control systems are simulated with unity feedback control of TRMS. Computer simulation of TRMS demonstrates improved results over the past contributions. The algorithm is superior over other robust control techniques because the singularities are taken care by the design itself. Following section gives the mathematical description of TRMS. Table 1 defines the abbreviations and symbolic presentations. All variables are measured in SI units.

Please cite this article as: P. K. Paul and J. Jacob, H2 Vs H1 control of TRMS via output error optimization augmenting sensor and control singularities2 Vs H1 control of TRMS via output error –>, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.07.001

3

P.K. Paul, J. Jacob / Ain Shams Engineering Journal xxx (xxxx) xxx Table 1 Nomenclature. Symbol

Definition

w

pitch angle from rest moment of inertia of the vertical rotor nonlinear static momentum in horizontal plane gravity momentum friction forces momentum in horizontal plane gyroscopic momentum in horizontal plane main motor torque main motor control input azimuth angle from rest moment of inertia of the horizontal rotor nonlinear static momentum in vertical plane friction forces momentum in vertical plane cross reaction momentum in vertical plane tail motor torque tail motor control input Input/Output Transfer Function Linear Fractional Transformation peak overshoot steady state error settling time Maximum allowable rotor input voltage

I1 M1 MFG MBw MG

sM

uM u I2 M2 MBu MR

sT

uT I/O T.F. LFT Mp Ess Tss Umax

2. Mathematical description of TRMS TRMS has strong cross coupling between main rotor and tail rotor dynamics. The two rotors are in mutually perpendicular planes with a comparatively faster dynamics as compared to the rigid body. The angular momentum of one rotor affects the angular momentum of the other. The gyroscopic effect of the main rotor has strong impact on the yaw angle, while that of the tail rotor has weak effect on the pitch angle of the TRMS as evident from its mathematical model stated in [1] and cited in the subsections that follow.

€ ¼ M 2  M Bu  M R I2  u

ð7Þ

M2 ¼ a2  sT 2 þ b2  sT

ð8Þ

_Þ MBu ¼ B1u  w_ þ B2u  signðu

ð9Þ

_ R ¼ kc T 0 s_ M þ kc sM  M R TpM

ð10Þ

T 21 s_ T ¼ T 20 sT þ k2 uT

ð11Þ

Different vectors used in Eqs. (1)–(11) are illustrated in Fig. 3 [1]. Table 2 presents the list of parameters and the nominal values as used in the present work. It is already stated that system modeling is an essentially primary step in any control system design problem. This paper will limit its scope to derivation of a linear nominal model for TRMS. The following section discusses modeling problem of TRMS in detail. It includes linearization of TRMS. In later sections generalized plant formulation using the linear nominal model of TRMS is discussed. 3. State space modeling of TRMS From the set of nonlinear mathematical equations of TRMS stated in Eqs. (1)–(11), the linear state space model may be obtained as:

X_ ¼ AX þ BU

ð12Þ

Y ¼ CX þ DU

ð13Þ

where the state vector is

X¼



w u

_ w_ u

i:e:; X ¼ ½ x1

x2

sM x3

x4

sT MR x5

x6



ð14Þ x7 

ð15Þ

The input vector is

2.1. Moments in the vertical plane

U ¼ ½ uM The nonlinear equations in time domain pertaining to the pitch movement of TRMS are quoted from Eqs. (1)–(6):

uT 

ð16Þ

The output vector is

€ ¼ M 1  M FG  MBw  M G I1  w

ð1Þ

Y ¼ ½w u

M 1 ¼ a1  sM 2 þ b1  sM

ð2Þ

To linearize the nonlinear terms in Eqs. (1)–(11), Taylor series is used and Jacobians are obtained.

T

ð17Þ

Main rotor of TRMS provides necessary thrust for pitch. Rest position of pitch is balanced by a counter weight, which is modeled by choice, to set the angle of elevation as desired. The elevation of TRMS functionally connects, gyroscopic momentum in horizontal plane to gravity momentum as depicted in Eq. (3). The weight and position of counter weight provide necessary inputs for designing the gravity compensation block, to be used in feed forward compensation, as mentioned in a later section.

M FG ¼ M G  sin ðwÞ M Bw


 ¼ B1w  w_ þ B2w  sign w_

ð3Þ ð4Þ

_  cos ðwÞ M G ¼ K gy  M1  u

ð5Þ

T 11 s_ M ¼ T 10 sM þ k1 uM

ð6Þ

2.2. Moments in the horizontal plane The nonlinear equations in time domain pertaining to the yaw movement of TRMS are quoted from Eqs. (7)–(11):

Fig. 3. TRMS working model [1].

Please cite this article as: P. K. Paul and J. Jacob, H2 Vs H1 control of TRMS via output error optimization augmenting sensor and control singularities2 Vs H1 control of TRMS via output error –>, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.07.001

4

P.K. Paul, J. Jacob / Ain Shams Engineering Journal xxx (xxxx) xxx

Table 2 Parameters and values [1]. Parameter

Value

I1 – moment of inertia of vertical rotor I2 – moment of inertia of horizontal rotor a1 – static characteristic parameter b1 - static characteristic parameter a2 – static characteristic parameter b2 – static characteristic parameter MG – gravity momentum B1w – friction momentum function parameter B2w – friction momentum function parameter B1u – friction momentum function parameter B2u – friction momentum function parameter Kgy – gyroscopic momentum parameter k1 – motor 1 gain k2 – motor 2 gain T11 – motor 1 denominator parameter T10 – motor 1 denominator parameter T21 – motor 2 denominator parameter T20 – motor 2 denominator parameter Tp – cross reaction momentum parameter T0 – cross reaction momentum parameter kc – cross reaction momentum gain

6.8  102 kg m2 2  102 kg m2 0.0135 0.0924 0.01 0.09 0.32 N-m 6  103N-m-s/rad. 1  103N-m-s/rad. 1  101N-m-s/rad. 1  103N-m-s/rad. 0.05 s/rad 1.1 0.8 1.2 1 1 1 2 3.5 0.2

2

0

0

6 0 0 6 6 4:348 0 6 6 A¼6 0 0 6 6 0 0 6 6 4 0 0 0

0

1

0

0

0 0:088

1 0

0

0 0 0

5 0 0

0

0

0 0 1:246

0

1:482 0:833 0

3:6 0 1

0:017

0

0

3

7 7 7 7 7 18:75 7 7 7 0 7 7 5 0 0 0

0:5 ð18Þ

 B¼  C¼  D¼

0 0

0 0 1

0 0

0 0

0 0 0

1 0

1 0

0 0 0

0 0

0 1

0 0 0

0 0

0 0

T

ð19Þ

 ð20Þ



0 0

ð21Þ

The 2  2 T.F. matrix of TRMS, derived from its state space model given in Eqs. (12)–(13), is presented as:



 g 11 ðsÞ g 12 ðsÞ g ðsÞ g 22 ðsÞ " 21 #  0 1:359= s3 þ 0:9091s2 þ 4:706s þ 4:278   ¼ ð17:5s þ 5Þ= s4 þ 1:409s3 þ 0:4545s2 3:6= s3 þ s2

GðsÞ ¼

ð22Þ This nominal model G(s) will be used in the formulation of the generalized plant described in the next section. 4. General H2/H‘ control configuration The general control configuration for designing H2/H1 controller is shown in Fig. 4. The designing of the controller K(s) starts with generalized plant formulation for the present problem. The input vectors W(s) and U(s) stand for the exogenous inputs (disturbances) and control inputs, respectively. The output vectors Z(s) and Y(s) are the constrained/penalized error signals and outputs, respectively. The generalized plant P(s) includes the actual plant G(s) along with all subsystems except the controller, K(s).

Fig. 4. General control configuration.

This configuration is confined to the linear time invariant MIMO finite order systems tested for its well-posedness with finite H2/H1 norms. Commonly available optimization methods for RHP zero plants frequently fail prematurely solely because of the divide by zero encounters while iteration, if not taken care at the design stage. Manipulating programming code or resetting word length of variable types in simulation software is rather an uneducated approach in solving such hurdles. A sophisticated output error optimization approach is proposed in this paper to overcome such hurdles. State space representation of P(s) is considered as

_ XðsÞ ¼ AXðsÞ þ B1 WðsÞ þ B2 UðsÞ

ð23Þ

ZðsÞ ¼ C 1 XðsÞ þ D11 WðsÞ þ D12 UðsÞ

ð24Þ

YðsÞ ¼ C 2 XðsÞ þ D21 WðsÞ þ D22 UðsÞ

ð25Þ

Thus, the generalized plant acquires the form in state space as:

2

A ½ B1 6  D PðsÞ ¼ 4 C 1 11 C2 D21

3 B2  7 D12 5

ð26Þ

D22

with inputs W(s) and U(s), outputs Z(s) and Y(s). The robust controller K(s), to be designed by H2/H1 optimization algorithm, is presented in state space as:

X_ k ðsÞ ¼ Ak X k ðsÞ þ Bk YðsÞ

ð27Þ

UðsÞ ¼ C k X k ðsÞ þ Dk YðsÞ

ð28Þ

The controller coefficients Ak, Bk, Ck and Dk are real valued matrices to be determined by the proposed output error optimization technique. 4.1. H2 and H1 optimization algorithm Objective of H2 and H1 optimization algorithm is to minimize the respective norms of the closed loop matrix Tzw(s) from W(s) to Z(s) for a given generalized plant P(s) [20]. While an H2 optimization algorithm can very well iterate to an optimal value of closed loop H2 norm (c < 1), the H1 optimization algorithm will compromise a strictly proper controller at the cost of optimal norm. In other words, a suboptimal solution for an H1 optimization algorithm will lead to designing a strictly proper controller with closed loop H1 norm > 1. Thus, the H2 optimization problem can be stated as finding a stabilizing controller K(s) that minimizes jjT ZW jj2 . Similarly, the H1 optimization problem may be stated as finding a stabilizing controller K(s) that minimizes jjT ZW jj1 such that 0 < jjT ZW jj1 < c as mentioned by Delgado and Zhou in [21]. An iteration algorithm results as closed loop gain less than c. Initially, limits of c are considered as clow and chigh assigning cn = chigh. If iteration completes, then



cnþ1 ¼ 0:5 cnlow þ cn n cnþ1 high ¼ c



ð29Þ ð30Þ

Please cite this article as: P. K. Paul and J. Jacob, H2 Vs H1 control of TRMS via output error optimization augmenting sensor and control singularities2 Vs H1 control of TRMS via output error –>, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.07.001

5

P.K. Paul, J. Jacob / Ain Shams Engineering Journal xxx (xxxx) xxx

early independent and the rows of the matrix C2(sI-A)1B1 + D21 are linearly independent.

Otherwise, if iteration fails, then




cnþ1 ¼ 0:5 cnhigh þ cn



ð31Þ

n cnþ1 low ¼ c

ð32Þ

The iteration repeats until we get,






cnhigh  cnlow < tolerance

ð33Þ

Stable H2/H1 controllers can be designed by directly computing the norm constrained stable transfer matrices Q in the H2/H1 suboptimal controller parameterization. First the problem is transformed into nonlinear unconstrained optimization problem. Then, a two stage numerical search is carried out. Consider Gs(s) as a stable transfer matrix with state space realization,

 Gs ðsÞ ¼

A

B

C

0

 ð34Þ

Then,

jjGs jj22





¼ TraceðB QBÞ ¼ TraceðCRC Þ

ð35Þ

Then, R and Q are the controllability and observability gramians, which can be obtained by solving the Lyapunov equations,

AR þ RA þ BB ¼ 0

ð36Þ

A Q þ QA þ C  C ¼ 0

ð37Þ

Consider the generalized plant P(s) is Eq. (26). Rewriting the same for the purpose of lower Linear Fractional Transformation (LFT),

2

½ B1 A 6C  D PðsÞ ¼ 4 1 11 C2 D21

3 B2   P 11 7 D12 5 ¼ P 21 D22

P12 P22

 ð38Þ

Recalling Fig. 4,

T zw ¼ F l ðP; K Þ ¼ P11 þ P21 KðI  P22 KÞ1 P 21

ð39Þ

The lower LFT presented in Eq. (39) is optimized to iterate optimal H2 controller and suboptimal H1 controllers. 4.2. Conditions for well-posedness A well-posed feedback loop establishes all transfer functions in the feedback loop as well defined and proper, failing which the optimization algorithm is prone to collapse [22–25]. The three conditions to be tested to assure well-posedness of the system are considered in the rest of this section. Testing for well-posedness is equivalent to validating the controllability and observability of the generalized plant formulated in the context.

4.2.3. Condition 3 In case of H2 optimization, the condition, D11 ¼ 0 and D22 ¼ 0 are treated separately. 4.2.3.1. Condition D11 ¼ 0. The software used in many cases do not support optimization with this condition. So, with such occurrence the existing algorithms may be modulated to acquire D11 –0. 4.2.3.2. Condition D22 ¼ 0. This case resembles to direct transfer of input to output. It is compensated manually. 5. Design of H2/H‘ controllers by output error optimization The desired performance of TRMS is specified in terms of a reference model G0(s) having the form as:



G 0 ð sÞ ¼



g 01 ðsÞ

0

0

g 02 ðsÞ

ð40Þ

g 01 ðsÞ and g 02 ðsÞ are chosen as standard second order systems satisfying the implied stability conditions. In the present technique, outputs of the generalized plant P(s) are penalized by comparing outputs of TRMS with that of G0(s). More specifically, output errors are penalized in this case. The reference plant, G0(s) generates desired outputs. Any deviation of actual outputs (pitch and yaw displacements of TRMS) is considered as controlled error outputs of the plant P(s) to be tackled by H2/H1 control algorithms. Further, solving H2 and H1 problems using mathematical tools are often limited by sensor and control singularities at zero and infinite frequencies, respectively, in addition to the deficit observable and/or controllable variables. To overcome this hurdle, control singularity at infinite frequency is compensated by adding an additional plant output as euV(s). Similarly, the sensor singularity at steady state is compensated by adding an additional disturbance term eyW(s). Unlike previous efforts to augment physical input/output variables to overcome singularity as well as controllability and/or observability issues, mathematical augmentation of such variables in generalized plant formulation is the essence of the approach proposed in this paper. Thus the generalized plant will have four controlled outputs, two sensor outputs, two reference inputs and two disturbance inputs. Fig. 5 is a line diagram presenting interconnection of subsystems. TRMS being a 2  2 plant, Fig. 5 can be illustrated more elaborately from physical perspective as shown in Fig. 6: T

Input vector; ½ W

U  ¼ ½ r1

r2

Output vector; ½ Z

Y T ¼ ½ Z 1

Z2

y1

y 2 T

¼ ½ q1

z1

q2

z2

w1

State vector; X ¼ ½ x1 x2 x3 x4 x5

w2

v 1 v 2 T y1

x6 x7 x8

y 2 T

ð42Þ

x9 x10 x11 T ð43Þ

4.2.1. Condition 1 This constraint pertains to the controllability and observability tests for the pairs (A, B2) and (C2, A), respectively. This is to ensure a stable feedback control system. Failing these tests, the system needs to be amended with additional control inputs and measurements for stability. 4.2.2. Condition 2 The second condition ensures that any exogenous (disturbance) input is measured at output and considered for generating control signal. Thus, the columns of the matrix C1(sI-A)1B2 + D12 are lin-

ð41Þ

Fig. 5. Control scheme of TRMS as output error optimization problem.

Please cite this article as: P. K. Paul and J. Jacob, H2 Vs H1 control of TRMS via output error optimization augmenting sensor and control singularities2 Vs H1 control of TRMS via output error –>, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.07.001

6

P.K. Paul, J. Jacob / Ain Shams Engineering Journal xxx (xxxx) xxx

 D12 ¼  D21 ¼  D22 ¼

0 0 0 0

e1u

1 0 0 1

0 0 0 0

0 0

0

0

T

ð53Þ

e2u 

ð54Þ

 ð55Þ

0 0

Thus,

2

Fig. 6. Elaborate control structure for robust output error optimization technique with TRMS.



_ i:e:; X ¼ w u w_ u

sM sT MR y10 y_ 10 y20 y_ 20

T

0 1 ð44Þ

 g 01 ðsÞ ¼ xn1 2 = s2 þ 2n1 xn1 s þ xn1 2

ð45Þ

 g 02 ðsÞ ¼ xn2 2 = s2 þ 2n2 xn2 s þ xn2 2

ð46Þ

The matrices in state space representation can be expressed as:

2 6 6 6 6 6 6 6 6 6 6 A¼6 6 6 6 6 6 6 6 6 4

0 0 0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

4:348 0

0 0 0 0 5

0 0

0 0

1:246 1:482

3:6

0 0 18:75

0 0

0:833 0

0 1

0 0



0 0

0 0 2n2 xn2

0:5 0

0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

xn1 2 0

2n1 xn1 0 0

0 0 0

0 0 0

0 0 1

0 0

0 0 0

1 0 0

0 0 0

0 1

T

0 0 0

0 0 0

0 0 0

0 0

0 0 0

1 0 0 0 1 0

0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

3 ð50Þ

0  0

 ð51Þ

ð52Þ 44

g 22 7 7 7 0 7 7 e2u 7 7 7 0 5

ð56Þ

g 22

xn2 2

0

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

ð47Þ

All the conditions given in Section 4.2 are tested for wellposedness of the generalized plant. Both the H2 and H1 controllers designed are of type zero and 11th order. A decoupler is suitable chosen to minimize the cross path interactions between yaw and pitch. Simulation results of pitch and yaw displacements are illustrated and discussed in rest of this section. Rotor inputs in simulation interface of MATLAB are limited to ±2.5 V. 6.1. H2 control simulation results Numerical values assigned to different parameters and constants for H2 optimization are as follows:

2

D11

3 0  0 . . . 6 . . .. 7 ¼ 4 .. 5

ð48Þ

ð49Þ

7 7 7 5

3

6. Results and discussion

0 0 0

6 0 1 0 0 0 0 0 0 0 1 0 6 C1 ¼ 6 4 0 0 0 0 0 0 0 0 0 0 0

C2 ¼

0 0

0 0

0 0

2

0 0

0:017 0

0 1 0

0

0

3

0 0

0 0

0 0 0

0

g 22 e2y

0 0

0 0

0 0

g 11

0 0

0 0

e2y

0

0 0

0 0

7 7 7 5

e1u

0 0

3T

0

0 0

0

0 1

0 0 g a1 xn1 2 0 0 0 0 0 0 g a2 xn2 2 0 0 0 0 0 0

g 22 e2y

0

0

g 11

0

The generalized model of P(s) as obtained in Eqs. (47)–(56) is used to iterate c using H1 and H2 optimization algorithms. An optimal H2 controller and a suboptimal H1 controller are thus obtained by iteration. The controllers are tested by simulating TRMS with main rotor feed-forward gravity compensation function as 0.3(w + 1). Simulation results are presented and discussed in the following section.

0

0 0 0 0 0 0 6 0 0 0 0 0 0 6 B1 ¼ 6 4 0 0 0 0 e1y 0 0 0 0 0 

0

0

2

B2 ¼

g 01 0 g 11 e1y 6 0 g 0 6 02 6 6 0 0 0 P ð sÞ ¼ 6 6 0 0 0 6 6 4 1 0 g 11 e1y

xn1 ¼ 0:5; xn2 ¼ 0:5; g a1 ¼ 1:02; g a2 ¼ 1:02; e1y ¼ 0:01; e2y ¼ 0:01; e1u ¼ 0:01; e2y ¼ 0:01; n1 ¼ 0:9; n2 ¼ 0:9; Nominal model of linearized TRMS obtained in Eq. (22) are quoted below for reference (see Figs. 7a and 7b, Figs. 8a and 8b).

  g 11 ðsÞ ¼ 1:359= s3 þ 0:9091s2 þ 4:706s þ 4:278 ;g 22 ðsÞ ¼ 3:6= s3 þ s2

Please cite this article as: P. K. Paul and J. Jacob, H2 Vs H1 control of TRMS via output error optimization augmenting sensor and control singularities2 Vs H1 control of TRMS via output error –>, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.07.001

P.K. Paul, J. Jacob / Ain Shams Engineering Journal xxx (xxxx) xxx

7

6.2. H1 control simulation results Numerical values assigned to different parameters and constants for H-infinity optimization are as follows:

xn1 ¼ 0:5; xn2 ¼ 1e  3; n1 ¼ 0:9; n2 ¼ 0:9; e1y ¼ 0:01; e2y ¼ 0:01; e1u ¼ 0:01; e2y ¼ 0:01; g a1 ¼ 1:02; g a2 ¼ 1e  2;

Fig. 7a. Pitch output against set point of 28° (without gravity compensation) – H2 control.

g 11 ðsÞ and g 22 ðsÞ are taken as given in Section 6.1 (see Figs. 9a and 9b, Figs. 10a and 10b). Yaw control is achieved within specifications with H1 controller as illustrated in the following Figs. 11a and 11b, Figs. 12a and 12b.

Fig. 9a. Pitch output against set point of 28° (without gravity compensation) – H1 control. Fig. 7b. Main rotor input for pitch set point of 28° (without gravity compensation) H2 control.

Fig. 8a. Pitch output against set point of 28° (with gravity compensation) – H2 control.

Fig. 8b. Main rotor input for pitch set point of 28° (with gravity compensation) H2 control.

Fig. 9b. Main rotor input for pitch set point of 28° (without gravity compensation) H1 control.

Fig. 10a. Pitch output against set point of 28° (with gravity compensation) – H1 control.

Please cite this article as: P. K. Paul and J. Jacob, H2 Vs H1 control of TRMS via output error optimization augmenting sensor and control singularities2 Vs H1 control of TRMS via output error –>, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.07.001

8

P.K. Paul, J. Jacob / Ain Shams Engineering Journal xxx (xxxx) xxx

Fig. 10b. Main rotor input for pitch set point of 28° (with gravity compensation) H1 control.

Fig. 12a. Yaw output against set point of 60° – H1 control.

Fig. 11a. Yaw output against set point of 10° – H1 control.

Fig. 12b. Tail rotor input for yaw set point of 60° – H1 control.

Table 3 Relative and absolute stavility parameters – H2 Vs H1 pitch control. Sl. No.

Parameter

H2 control

H-infinity control

Remarks

1

Peak overshoot

3.57%

2.3%

2

Peak overshoot

10.35%

9.3%

3

Settling time (2% margin)

11 s

10.5 s

4

Settling time (2% margin)

11 s

11.5 s

5

Steady state error

0.89%

0.357%

6

Steady state error

0.89%

0

Without gravity compensation With gravity compensation Without gravity compensation With gravity compensation Without gravity compensation With gravity compensation

Fig. 11b. Tail rotor input for yaw set point of 10° – H1 control.

Table 3 contains the list of relative and absolute stability parameters for (1) H2 controller and (2) H1 controller: Table 3 summarizes a range of stability parameters of TRMS with H2 and H1 controllers for pitch. It can be concluded that H1 controller exhibits superior results in aggregate. However, performance of H1 controller without gravity term is not upto the mark while it performs well in the presence of gravity compensation. In addition, H2 optimization did not yield satisfactory results for yaw displacement whereas with H1 controller desirable yaw movements are observed. Further, the approach proposed in this work imbibes the relative and absolute stability requirements in the form of second order transfer functions (G0(s)). Thus, this method is superior to

the sensitivity function shaping methods in which the controlled outputs are penalized by using a weighting function which is again optimized by trial and error method. Hence, output error optimization technique is a straight forward method and minimizes controller design time and other requirements. 7. Conclusion The present work establishes the superiority of (1) H1 controller over H2 controller when optimized via the proposed output error optimization technique tested on TRMS and (2) output error optimization technique over sensitivity function shaping method using weighting functions. Thus, output error optimization

Please cite this article as: P. K. Paul and J. Jacob, H2 Vs H1 control of TRMS via output error optimization augmenting sensor and control singularities2 Vs H1 control of TRMS via output error –>, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.07.001

P.K. Paul, J. Jacob / Ain Shams Engineering Journal xxx (xxxx) xxx

technique proposed in this paper is time efficient and guaranteed approach. As order of the H1 controller K(s) and that of the plant P(s) are the same, the controller is strictly proper (numerator – 10th order and denominator – 11th order) guaranteeing strong stability of control loop. The work contributes to the class of 2 by 2 plants with non-minimum phase dynamics like TRMS that cannot be tackled using traditional approaches. This approach clearly explains the compensation of singularities at zero and infinite frequencies which are common hurdles in computer simulations. It further guarantees controllability and observability via wellposedness tests. Simulation results have illustrated the robust behavior of the controlled system as it is clearly shown that the controllers are designed using nominal model of TRMS, reference model and generalized plant model. Based on the simulation results, H1 controller is established as the superior one in this case. The present method can be extended to control the class of systems where number of measurements is deficit for full control of the plant. Results of the real time implementation of this controller will be reported in an ensuing paper. Acknowledgements

[14]

[15]

[16] [17]

[18]

[19] [20]

[21] [22] [23]

This research work was performed at the Control Systems laboratory of the Electrical Engineering Department, National Institute of Technology Calicut, India. The authors are thankful to the technical staff of the Institute for providing the experimental facilities and arrangement. The authors appreciate the valuable comments from the reviewers. References [1] Feedback Instruments Ltd. Twin Rotor MIMO System Control Experiments 33949S Laboratory Manual, U.K.; 2005. [2] Martinez ML, Viva C, Ortega MG. A multivariable nonlinear H1 controller for a laboratory helicopter. In: Proceedings of the 44th IEEE Conference on Decision and Control; 2005. p. 4065–4070. [3] Martinez ML, Vivas C, Ortega MG, Rubio FR. Nonlinear L2 control of a laboratory helicopter with variable speed rotors. Automatica 2007;43:655–61. [4] Smerlas AJ, Walker DJ, Postlethwaite I, Strange ME, Howitt J, Gubbels AW. Evaluating H1 controllers on the NRC Bell 205 fly-by-wire helicopter. Control Eng Pract 2001;9:1–10. [5] Vilchis JCA, Brogliato B, Dzul A, Lozano R. Nonlinear modelling and control of helicopters. Automatica 2003;39:1583–96. [6] Mullhaupt Ph, Srinivasan B, Levine J, Bonvin D. Cascade control of the toycopter. In: Proceedings of European Control Conference (ECC’99), Karlsruhe; 1999. [7] Lu TW, Wen P. Time optimal and robust control of twin rotor system. IEEE International Conference on Control and Automation ThA3-2; 2007. p. 862– 866. [8] Skogestad S, Postlethwaite I. Multivariable feedback control - analysis and design. 2nd ed. England: John Wiley and Sons; 2005. [9] Nejjari F, Rotondo D, Puig V, Innocenti M. Quasi-LPV modelling and non-linear Identification of a twin rotor system. In: 20th Mediterranean Conference on Control & Automation (MED) Barcelona, Spain; 2012. p. 229–234. [10] Ahmad SM, Chipperfield AJ, Tohki MO. Modelling and control of a twin rotor multi-input multi- output system. In: Proceedings of American Control Conference; 2000. p. 1720–1724. [11] Ahmad SM, Chipperfield AJ, Tohki MO. Dynamic modelling and optimal control of a twin rotor mimo system. In: Proceedings of IEEE; 2000. p. 391–398. [12] Ahmad SM, Chipperfield AJ, Tohki MO. Dynamic modelling and linear quadratic Gaussian control of a twin-rotor multi-input multi-output system. In: Proceedings of Instn. Mech. Engrs, Journal of Systems and Control Engineering 217 Part I; 2003. p. 203–227. [13] Ahmad SM, Chipperfield AJ, Tohki MO. Dynamic modelling and open-loop control of a two-degree-of-freedom twin-rotor multi-input multi-output

[24] [25]

9

system. In: Proceedings of Instn. Mech. Engrs, Journal of Systems and Control Engineering, 218 Part I; 2004. p. 451–463. Belkheiri M, Rabhi A, Boudjema F, Elhajjaji A, Bosche J. Model parameter Identification and nonlinear control of a twin rotor mimo system – trms. In: Proceedings of the 15th IFAC Symposium on System Identification, Saint-Malo, France; 2009. p. 1487–1492. Tanaka H, Ohta Y, Okimura Y. A local approach to lpv-identification of a twin rotor mimo system. In: Proceedings of the 18th World Congress - The International Federation of Automatic Control, Milano (Italy); 2011. p. 7749– 7754. Rahideha A, Shaheed MH, Huijberts HJC. Dynamic modelling of a TRMS using analytical and empirical approaches. Control Eng Pract 2008;16:241–59. Ekbote AK, Srinivasan NS, Mahindrakar AD. Terminal sliding mode control of a twin rotor multiple-input multiple-output system. In: Proceedings of the 18th World Congress - The International Federation of Automatic Control, Milano (Italy); 2011. p. 10952–10957. Vishnupriyan J, Manoharan PS, Ramalakshmi APS. Uncertainty modeling of nonlinear 2-dof helicopter model. In: International Conference on Computer Communication and Informatics (ICCCI -2014), Coimbatore, India; 2014. Chalupa P, Prˇikryl J, Novák J. Modelling of twin rotor MIMO system. Procedia Eng 2015;100:249–58. Megretski A. On the order of optimal controllers in the mixed H-2/H-infinity control. In: Proceedings of the 33rd conference on Design and Control, Lake Buena Vista (Florida); 1994. p. 3173–3174. Delgado DUC, Zhou K. A parametric optimization approach to H1 and H2 strong stabilization. Automatica 2003;39:1205–11. Zhou K, Doyle JC, Glover K. Robust and optimal control. New Jersey: Prentice Hall; 1996. Iwasaki T, Hara S. Well-posedness of feedback systems: insights into exact robustness analysis and approximate computations. IEEE Trans Autom Control 1998;43(5):619–30. Francis BA. A course in H-infinity control theory, lecture notes in control and information science. Berlin: Springer-Verlag; 1987. Doyle JC, Francis BA, Tannenbaum AR. Feedback control theory. New York: Macmillan; 1992.

Parthish Kumar Paul is Assistant Professor with the Department of Electrical and Electronics Engineering, Dr. Akhilesh Das Gupta Institute of Technology and Management, New Delhi. He received B.E. in Electrical Engineering from Regional Engineering College, Durgapur, India in the year of 1999, M. Tech. in Electrical Engineering with specialization in Instrumentation and Control Systems in the year of 2008 and PhD in Electrical Engineering in the year of 2019 from National Institute of Technology, Calicut, India. His research interest is in the area of robust control of MIMO systems. He has special interest in linear and nonlinear control theory, system modeling and circuit theory. He is member of IEEE, IIIE and ISTE. The author is developing site oriented text books for Elctricians. He is nterested in UAV/UAS for further research.

Jeevamma Jacob is Professor with the Department of Electrical Engineering, National Institute of Technology, Calicut, Kerala, India. She received the B.Tech degree in Electrical Engineering in 1983 from the University of Kerala, India and M. Tech. (Instrumentation & Control) degree in 1985 from National Institute of Technology, Calicut (formerly REC, Calicut). She received the Ph.D (Control Systems) degree in 1994 from Indian Institute of Technology, Bombay, India. Her areas of interest include robust multivariable controller design aspects extended to the areas of power systems, process control, digital control and biomedical engineering. She has published several papers in her areas of interests. Dr. Jeevamma Jacob is member of IEEE, ISTE and SSI.

Please cite this article as: P. K. Paul and J. Jacob, H2 Vs H1 control of TRMS via output error optimization augmenting sensor and control singularities2 Vs H1 control of TRMS via output error –>, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.07.001