Model predictive control for twin rotor MIMO system (TRMS)

A Twin Rotor MIMO System (TRMS) is an aerodynamic experimental system with high nonlinearity which includes two inputs, two outputs, and six states. In this paper we will present the results from the application of Model Predictive Control (MPC) for TRMS based on its mathematical model we have built recently.

ISSN 1859-1531 - THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 12(85).2014, VOL 39 MODEL PREDICTIVE CONTROL FOR TWIN ROTOR MIMO SYSTEM (TRMS) Nguyen Thi Mai Huong, Mai Trung Thai, Nguyen Huu Chinh, Tran Thien Dung, Lai Khac Lai Thai Nguyen University of Technology Abstract - A Twin Rotor MIMO System (TRMS) is an aerodynamic experimental system with high nonlinearity which includes two inputs, two outputs, and six states In the world, this system has been studied and applied in reality in order to evaluate and implement the advanced control algorithms [1], [2], [3], [8], [9] In Vietnam, although the TRMSs have been installed in some university laboratories, it is still difficult to use them for testing modern control algorithms because there is no exact mathematical model of the system The documents and software provided on a laboratory equipment provider in the algorithm are confined to the classical PID controller In this paper we will present the results from the application of Model Predictive Control (MPC) for TRMS based on its mathematical model we have built recently [12] Key words - Model Predictive Control (MPC); State parametters; Twin rotor MIMO system (TRMS); cross-coupling channels; yaw angle (horizontal angle); pitch angle (vertical angle Introduction MPC is one of the advanced control techniques suitable for the problems of controlling industrial processes The construction of the predictive model built on complex domain as GPC (General Predictive Control), or an equivalent as DMC (Dynamic Matrix Control) is the most suitable for SISO objects [10], [11] The TRMS is a MIMO and a nonlinear system, therefore constructing predictive models is performed in the time domain because it is easy to linearize and calculate Construction of Methodology for MPC algorithms Consider a nonlinear system with nu inputs, nx outputs and ny states are described as the state space equations below: x(k + 1) = f ( x(k ), u (k )) (1) y(k ) = h( x(k ))  Where x(k) is the state vector, u(k) is the input vector, and y(k) is the output vector, all at instant k It can be linearised adaptively at each real sample time k (In model predictive control, two sample instants are considered and should be clarified to prevent from misunderstanding One is the real sample time, and the other is the internal sample time In term u(k + i k ) , k is the real sample time and k + i is the internal sample time) as the state equations of the discrete space below: x(k + 1) = A(k ) x(k ) + B(k )u (k ) (2) y ( k ) = C (k ) x (k ) or can be represented by a combination of state- dependent state-space equations as: x(k + 1) = A( x(k )) x(k ) + B( x(k ))u (k ) (3) y(k ) = C ( x(k )) x(k )   The state variables and the inputs related to previous instant are used as initial conditions to linearise the non-linear system at each time Making linearized nonlinear system Np times at each sampling instance adaptively according to Np operating points from earlier periods of the optimum result:  xˆ ( k + i + k ) = A( x ( k + i k )) xˆ ( k + i k ) + B ( x ( k + i k ))uˆ ( k + i k ) yˆ ( k + i k ) = C ( x ( k + i k )) xˆ ( k + i k ) (4) i = 0,1, , N p − In order to simplify the representation of the equations, the state dependent matrix A( x(k + i k )) is shown as A(k + i k ) and similar are the other state-dependent matrices To find the linear models, one can use the known values of x(k + i k − 1) instead of the unknown x(k + i k ) , where i = 0, 1, …, Np – In order to solve the optimization problem of the MPC, and obtain the relationship between the internal model outputs during the prediction horizon interval, 1≤ i ≤ NP, and the internal model inputs during the control horizon interval, 1≤ i ≤ NC, where Np and Nc are the prediction and control horizons If the relationship is linear and the constraints are also linear, there is an optimization problem in quadratic form In the prediction horizon, the state vector can be expressed in terms of the state available vector x(k) and the future input vectors:  i  xˆ(k + i + k ) =  A(k + i − j k )  x(k )  j =0  (5) i   i − n −1    +     A(k + i − j k )  B(k + n k ) uˆ (k + n k )  n=0     j =0 It is common to use the input difference between two consecutive instants, uˆ(k + i k ), instead of the input itself, uˆ(k + i k ), using uˆ (k + i k ) = uˆ (k + i k ) − uˆ (k + i − k ) [5] The only input changes during rest-of-control and did not change after, namely uˆ (k + i k ) = uˆ (k + N C − k ) this means that uˆ(k + i k ) = 0for Nc ≤ i ≤ Np-1 The input vectors related to the reference input vector: j uˆ (k + j k ) = u (k − 1) +  uˆ (k + i k ) j = 0,1, , N C − (6) i =0 Subsituting equation (6) into equation (5) we obtain: xˆ ( k + i + k )    A( k + i − j k )  x ( k )   +   A( k + i − j k ) B ( k + n k ) u ( k − 1)      +     A( k + i − j k ) B ( k + n k )     i =   j =0 i n=0 i − n −1 j =0 min( i , N C − 1) m=0 i n=m  i − n −1 j =0  (7)   uˆ ( k + m k )  i = 0, , N p − The predicted outputs are represented as: yˆ (k + i k ) = C (k + i k ) xˆ (k + i k ) + dˆ (k + i k ), i = 1, , N p (8) 40 Nguyen Thi Mai Huong, Mai Trung Thai, Nguyen Huu Chinh, Tran Thien Dung, Lai Khac Lai where dˆ   ny x1is the disturbance Subsituting equation (7) into equation (8) we obtain: Y (k ) = M C (k ) M A (k ) x(k ) + M C (k ) M B ( k )u ( k − 1) (9) + M C (k ) M U (k ) U ( k ) + M d ( k ) In which the matrix /vector: Y (k )  ny N p x1 , M C (k )  ny N p xnx N p , M A (k )  nx N p xnx , M B (k )  nx N p xnu , M U (k )  nx N p xnu NC , U (k )  nu NC x1 , M d (k )  ny N p x1 Objective function Suppose that the following objective function minimization as the constraint conditions (11) to (13): + iav: Armature current of the main motor (A); ωv: Rotational velocity of main rotor(rad/s); Sv: Angular velocity of TRMS beam in the vertical plane without affect of the tail rotor (rad/s) v:Vertical position (pitch angle) of the TRMS beam (rad) Uh: Input voltage signal of the tailmotor (V) Uv: Input voltage signal of the main motor (V) Tail rotor z Pivot beam   r (k + i ) − yˆ (k + i k )  (i )  r (k + i ) − yˆ (k + i k )  i =1 NC Sh: Angular velocity of TRMS beam in the horizontal plane without affect of the main rotor (rad/s); T NP J (k ) = ωh: Rotational velocity of the tail rotor (rad/s); P3 (10) T   uˆ (k + i − k )   (i)  uˆ (k + i − k )  (11) (12) (13) Main rotor P1 P2 Counter balance beam −v Where r: Reference trajectory with dimension (ny x 1); δ: The weight matrix of tracking errors with dimension (ny x O2 ny); rx( R1 ) T (14) x Subsituting equation (9) into equation (14) the objective function is a quadratic form: J (k ) = U T (k ) H (k )U (k ) (15) +U T (k )G(k ) + c(k ) where H (k ) = 2( M UT (k ) M CT (k )QM C (k ) M U (k ) + R ) G (k ) = −2 M UT (k ) M CT (k )QE (k ) c(k ) = E T (k )QE (k ) E (k ) = M r (k ) − M C (k ) M A (k ) x(k ) − M C (k ) M B (k )u (k − 1) − M d (k ) TRMS Objects The proposed multistep Newton-type MPC based on the state - dependent is implemented on the TRMS, Figure The control objective is to control the yaw and the pitch angles (h, v) as accurate as possible The state variables, the input and output vectors of TRMS are as follows: x ( k ) = iah ( k ) h ( k ) S h ( k )  h ( k ) (16) iav (k ) v (k ) Sv (k )  v (k )  T (17) u(k ) = U h (k ) U v (k ) T y(k ) =  h (k ) v (k ) T Where: iah: Armature current of the tail motor (A); (18) ry(R1) y λ: The weight matrix of control efforts with dimension (nu x nu) The objective function can be written as: J (k ) =  M r (k ) − Y (k )  Q  M r (k ) − Y (k )  +U T (k ) RU (k ) Free beam −h O1 i =1 ymin  yˆ(k + i k )  ymax , i = 1, 2, , N p umin  uˆ(k + i − k )  umax , i = 1, 2, , NC umin  uˆ(k + i − k )  umax , i = 1, 2, , NC O3 P'1 Figure TRMS Model The nonlinear continuous state space equations of the TRMS are expressed in [8]: Rah kah h   − iah − h + f (U h )   L L L ah ah ah iah    kah h Btr f1 (h )     i  ah − h − h    J tr J tr J tr     lt f (h ) cos  v − f ( h ) − f3 ( h )     2 S h    D cos v + E sin  v + F     k mv cos  v  Sh +  h   2  D cos  + E sin  + F v v     R k  d   av av v  (19) iav =  − iav − v + f8 (U v )   Lav Lav Lav dt       k B f av  v mr (v )  v   iav − v −  J mr J mr J mr      Sv   f5 (v )(lm + k g  h cos  v ) − f9 ( v )      J v     + g ( A − B ) cos  − C sin  − 0.5  H sin    v v h v      v   Jv      kt S v + h   Jv   where Rah , Lah , kahh , J tr , Btr , lt , D, E , F , km , Rav , Lav , kavv , J mr , Bmr , lm , k g , g , A, B, C , H , J v , kt is the positive constant, h and v is defined as ISSN 1859-1531 - THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 12(85).2014, VOL   −  −  − + f (U )     J R J J J R     l f ( ) cos  − f (  ) − f ( )   S   D cos  + E sin  + F     k  cos     S +      D cos  + E sin  + F     d ( k  ) B f (  ) k    (22)   = −  −  − + f (U )  dt    J R J J J R   S   f ( )(l + k  cos  ) − f ( )   +    J      g ( A − B ) cos  − C sin  − 0.5  H sin          J     k     S +    J Although this reduced-order model does not affect the accuracy of the model, it can significantly affect the boot capacity calculations that reduce processor load and the speed of the optimization problem The nonlinear statespace equation above can be approximated and represented as a state space equation follows: x = A( x) x + Bu ( k ah h ) f1 (h ) Btr h h tr h ah tr t tr h v tr h h h ah 0.5 Alphav Reference 0.4 0.3 0.2 h v -0.1 -0.2 v m 0.1 h v h k ah h responses of Yaw angle and pitch angle track the reference in predictive window Especially, the cross-coupling channels between Yaw angle and pitch angle is best known As soon as h varies, v changes and vice versa Then the outputs track the inputs Pitch angle - alphav(rad) kmv cos  v (20) D cos2 v + E sin  v + F kt h v = Sv + (21) Jv f1 to f9 is the nonlinear functions When Lah