Design and Implementation of Proportional Integral Observer based Linear Model Predictive Controller

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This paper presents an interior-point method (IPM)
based quadratic programming (QP) solver for the solution of
optimal control problem in linear model predictive control
(MPC). LU factorization is used to solve the system of linear
equations efficiently at each iteration of IPM, which renders
faster execution of QP solver. The controller requires internal
states of the system. To address this issue, a Proportional
Integral Observer (PIO) is designed, which estimates the state
vector, as well as the uncertainties in an integrated manner.
MPC uses the states estimated by PIO, and the effect of
uncertainty is compensated by augmenting MPC with PIOestimated
uncertainties and external disturbances. The
approach is demonstrated practically by applying MPC to QET
DC servomotor for position control application. The proposed
method is compared with classical control strategy-PID
control.

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Design and Implementation of Proportional Integral Observer based Linear Model Predictive Controller

  1. 1. Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013 Design and Implementation of Proportional Integral Observer based Linear Model Predictive Controller Vihangkumar V. Naik, D. N. Sonawane, Deepak D. Ingole, Divyesh L. Ginoya and Vedika V. Patki Department of Instrumentation & Control Engineering, College of Engineering, Pune- 411 005 Maharashtra, India Email: naikvihang@gmail.com, dns.instru@coep.ac.in, dingole21@yahoo.in, dlginoya007@gmail.com, patki.vedika@gmail.comAbstract—This paper presents an interior-point method (IPM) fast response time and/or embedded applications wherebased quadratic programming (QP) solver for the solution of computational resource may be limited [4].optimal control problem in linear model predictive control Recently, many reports in the literature address applying(MPC). LU factorization is used to solve the system of linear MPC to control applications with short sampling intervals,equations efficiently at each iteration of IPM, which renders by adapting fast optimization algorithms. These algorithmsfaster execution of QP solver. The controller requires internalstates of the system. To address this issue, a Proportional solve the resulting QP by exploiting the special structure ofIntegral Observer (PIO) is designed, which estimates the state the control problem the MPC sub problem. IPM approachesvector, as well as the uncertainties in an integrated manner. the solution of Karush-Kuhn-Tucker (KKT) equations byMPC uses the states estimated by PIO, and the effect of successive decent steps. Each decent step is Newton’s likeuncertainty is compensated by augmenting MPC with PIO- step and the solution is obtained by solving system of linearestimated uncertainties and external disturbances. The equations using appropriate numerical methods in order toapproach is demonstrated practically by applying MPC to QET determine search direction. Matrix factorization approachDC servomotor for position control application. The proposed provides a means to simplify the computation involved inmethod is compared with classical control strategy-PID linear solver. Different matrix factorization methods are usedcontrol. such as Gauss elimination, QR, LU, and Cholesky [3], [5], [6].Index Terms—Model Predictive Control, Interior-Point Model predictive control has received considerableMethod, LU factorization, DC Servomotor, Proportional attention driven largely by its ability to handle hardIntegral Observer. constraints. An inherent problem is that model predictive control normally requires full knowledge of the internal states. I. INTRODUCTION In the physical implementation of control strategy based on the system states, if the states are not available for This paper considers Interior-point algorithm used on- measurement then state feedback will not be theline with Linear Model Predictive Control (MPC). Linear MPC implementable.assumes a linear system model, linear inequality constraints The problem of robust control system design with anand a convex quadratic cost function [1]. MPC can be observer based on the knowledge of input and output of theformulated as a quadratic programming (QP) problem and plant has attracted many researchers now-a-days [7], [8], [9],solved at each sampling interval. It thus has the natural ability [10], [11], [12], [20]. The observer proves to be useful in notto handle physical constraints arising in industrial only system estimation and regulation but also identifyingapplications [2]. failures in dynamic systems. This requirement of obtaining At each sampling instance MPC solves an online QP the estimates of uncertainty, as well as state vector in anoptimization problem, computes the sequence of optimal integrated manner is fulfilled by PIO. PIO approach basicallycurrent and future control inputs by minimizing the difference works when the system uncertainty varies slowly with respectbetween set-points and future outputs predicted from a given to time. It is an observer in which an additional term, which isplant model over a finite horizon in forward time. Then, only proportional to the integral of the output estimation error, isthe current optimal input is applied to the plant. The updated added in order to estimate the uncertainties and improve theplant information is used to formulate and solve a new optimal system performance [7], [8], [9].control problem at the next sampling instance. This procedure In this paper, we present an interior-point algorithm tois repeated at the each sampling instance and the concept is solve MPC problem, which utilizes Mehrotra’s predictor-called as receding horizon control MPC strategy. Since this corrector algorithm [13], and the linear system at each IPMquadratic program can be large depending upon control iteration is solved by LU factorization based Linear Solver.problem, MPC requires long computation times at each PIO is designed for state and uncertainties estimation. Thesampling instants, therefore it is usually restricted to systems remainder of the paper is organized as follows: section IIwith slow dynamics and large sampling intervals, such as describes MPC problem formulation. Section III presents thechemical processes [3]. The ability to solve the QP problem interior-point method. In section IV, the PIO theory is brieflyonline become critical when applying MPC to systems with© 2013 ACEEE 23DOI: 01.IJCSI.4.1.1064
  2. 2. Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013reviewed and applied for state and uncertainty estimation. where, N u is control horizon. The goal of the controller is toMathematical modeling of DC servomotor is derived inSection V. IPM QP solver results, the effectiveness of observer ˆ make the difference between the output, y (t  N P ) and thebased MPC scheme with simulation results is presented in reference, yref (t  N P ) as small as possible. This can besection VI. MPC implementation on Quanser QET DCservomotor is presented in section VII. defined by using a standard least squares problem. The objective function is defined as, II. MPC PROBLEM FORMULATION 1 NP 2 1 Nu 1 2 A discrete time linear time-invariant model of a system in J  2 i1 y(t  i)  yref (t  i)   u(t  i) R . ˆ Q 2 i1 (7)a state space form is given as, where, u (t  i )  u (t  i )  u (t  i  1) x(t  1)  Ax(t )  Bu(t ) . subjected to linear inequality constraints on system inputs, (1) y (t )  Cx(t )  umin  u (t )  umax where, y (t )  are output vector, u (t )  , where, yref is the set-point and while Q and R are theare input vector and x (t )  are internal states vector. . positive definite weight matrices [14]. To get optimal solution, A and B are system parameter matrices. Given a predicted the MPC problem can be formulated as a standard quadraticinput sequence, the corresponding sequence of state programming (QP) problem as,predictions is generated by simulating the model forward 1  J (U )   U T HU  f T U  subjected to AU  b  0. (8)over the prediction horizon ( N P ) intervals. 2  x(t  2)  Ax(t  1)  Bu (t  1)  where, H is (lN u  lN u ) Hessian matrix, f is 2 . (2)x(t  2)  A x(t )  ABu(t )  Bu (t  1) (lN u 1) column vector [14], [16]. N P 1x(t  N P )  A N P x(t )   j 0 ( A N P 1 j B )u ( j ). (3) III. INTERIOR-POINT METHODThe prediction model is given by, Consider the following standard QP problem, ˆ y (t  N P )  Cx (t  N P ). (4) min 1 xT Qx CT x subjected 2 to Ax  b  0, Lx  k  0 . xPutting the value of x (t  N P ) in (4), we get, (9) N 1y(t  N P )  C[ A N P x(0)   j P0 ( A N P 1 j B)u ( j )].ˆ Except equality constraints, (9) resembles (8). The KKT Conditions for optimality are given by, (5)In matrix form (5) is written as, Qx  AT y  LT z  C  Y  x(0)  U . (6) with, Ax  b  0   Lx  s  k  0 .Y   y (t  1) y(t  2)  y (t  N P ) , T ˆ ˆ ˆ T  (10) z s0 U  u (t ) u (t  1)  y (t  N u  1)  , T ˆ z, s  0    CA  where, y and z represents the Lagrange multipliers for  CA 2  equality and inequality constraints respectively. The slack  , variable s is introduced for converting inequality into     NP  equality. By applying Newton’s method, search direction CA  ( x,y ,z , s ) is obtained as shown in (11), where J ( x , y , z , s ) denotes Jacobian of f ( x, y, z , s ) ,  CB 0  0  CAB CB  0  x   r1    y        0  N p 1 N p 2 N p Nu  J ( x, y, z , s )      r2 . CA B CA B  CA B  z   r3  (11)     24  s   r4 © 2013 ACEEEDOI: 01.IJCSI.4.1.1064
  3. 3. Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013Considering only inequality constraints, augmented form of A  LU ;original system is given as, Solve LY  B; //forward substitution Q LT  x  r1   1      1  . (12) UX  Y ; //backward substitution  L  Z S   z  r3  Z r4  X   X 1; X 2 ; s  Z 1 (r4  Sz). (13) Interior-point method(IPM) algorithmWe use Mehrotra’s Predictor-corrector algorithm [13] to solve Initialization and input(12) and (13) for ( x ,  z ,  s ). Predictor or affine and X 0  0 0  0 ; Tcorrector-centering steps uses ( r1 , r3 , r4 ) values from (14) Z0  1 1  1 ; S0  1 1  1 ; Z0 , S0  0; T Tand (15) respectively to form the right-hand side of (12) and(13) which in turn finds ( x aff ,  z aff , s aff ) and e  1 1  1 ;  0.25 T(x cc , z cc , s cc ) [1], [15], [17], [18], [19]. Input Q, C , L, k matrices from given QP problem T  r1  Qx  C  L z T Z 0 S0  r     Lx  s  k   ; m- no. of constraints  3  . (14) m r4     ZSe  Start the loop and terminate if stopping criterion are satisfied   Predictor or affine step S  diag ( S 0 ); Z  diag ( Z 0 );  r1   0  r1aff  QX 0  C  LT Z 0 ; r3aff  LX 0  S0  k ; r     (15)  0 .  3 r4    diag (z aff )diag (s aff )e  e r4 aff  ZSe;   Solve (13), (12) by linear solver and computewhere,  is centering parameter and  is complementaritymeasure or duality gap. Thus, solution of linear system in x aff , z aff , s aff(13) is to be calculated twice at each iteration of IPM, whichputs major computational load on overall performance of IPM, Compute centring parameter and eventually on MPC. Linear system in (13) uses twodifferent right-hand side components but the same coefficient ( Z 0  z aff )T ( S 0  saff )matrix, so just one factorization of this matrix is required per  aff  ;iteration. Such system can be solved using Gauss elimination, mQR, LU, Cholesky factorization methods. 3 The coefficient matrix in a linear system (13) involved in      aff   ; IPM is symmetric and indefinite so conventional and more  numerically stable Cholesky factorization cannot be used tosolve the linear system. Compared to computational cost of Corrector and centring step to obtain search directionQR decomposition (2/3n3) and the problem of associated Z aff  diag ( z aff ); S aff  diag ( saff );pivoting with ill condition of Gauss elimination, LUfactorization proves to be more effective and accurate to solve r1cc  0; r3cc  0; r4 aff  Z aff S aff e   e;system of linear equations [5], [6]. We propose LUfactorization based Linear Solver for the same. The pseudo Solve (13), (12) by linear solver and computecode of LU factorization Linear Solver for (13) is given asfollows: x cc , z cc , s ccPseudo Code-LU factorizationA   A11 A12 ; A21 A22 ; Update ( x ,  z ,  s ) aswhere, A11  Q; A12  LT ; A22  L; A22   Z 1S (x, z, s)  ( x aff , z aff , s aff )  (x cc , z cc , s cc )B  B11; B22 ; Update X 0 , Z 0 , S 0 aswhere, B11  r1 , B22  ( r3  Z 1r4 ); X 0  X 0   x ; Z 0  Z 0   z ; S 0  S 0   s ; End of loop© 2013 ACEEE 25DOI: 01.IJCSI.4.1.1064
  4. 4. Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013 IV. OBSERVER DESIGN Simplifying (13) The implementation of MPC requires the measurements  ~(t  1)   A B  L  x   ~ (t ) xof all the states. As only an encoder is used for measurement     C 0  ~ .  ~ (t  1)  0 0  M  e (t ) of the motor shaft position in practice. PIO is designed in this e       section for present application. Consider a linear time-invariant system It follows that x(t ) is an estimate of x(t ) if and only if, ˆ q R   0 for q  1, , ( n  p ) . The generalized block~ (t  1)  A~ (t )  Bu (t )  L( y (t )  y (t ))  Be(t ) x x ˆ ˆ diagram of the observer based MPC system is shown in ˆ ˆ y (t )  Cx(t )  .  Fig. 1. ˆ ˆ ˆ e(t  1)  e(t )  M ( y(t )  y (t ))   V. MATHEMATICAL MODELLING OF DC SERVOMOTOR For mathematical model derivation of DC servomotor, thewhere ~ (t ) is the estimated state vector and e (t ) is the x ~ electrical & torque characteristic equations are formulated.estimation of uncertainties of p-dimensional vector. Lproportional and M integral observer gain matrices ofappropriate dimension. The system described by Eq. (16) is said to be a full-orderPI observer for system (1) if and only if lim ~ (t )  0 x  t  lim e~ (t )  0 .  t   where, ~(t )  x(t )  x(t ) represents the observer’s state x ˆ ~ ˆestimation error and e (t )  e(t )  e(t ) represents the errorin uncertainty estimation. We are assuming the following for derivation. Figure 2. Schematic diagram of DC Servomotor1. Matrices A, B, C are known. All uncertainties associated Fig. 2, shows the schematic diagram of armature controlled DC servomotor. Using Kirchhoff’s voltage law (KVL),with the matrices A and B are lumped into e(t ) . 2. The pair ( A, B) is controllable. Vm  R m I m  Lm   I m  k m  m . .   t 3. The pair ( A, C ) is observable. where, k m is the back-emf constant, R m is the motor winding resistance and L m is the motor winding inductance, I m is the current through the motor winding. TABLE I. NOMINAL DC SERVOMOTOR PARAMETERS [21] Description Symbol Value Unit Motor electric time constant e 7.74 * 10-5 s Moment of inertia J eq 2.21*10-5 Kg.m2 Motor maximum velocity umax 298.8 rad/s Motor maximum current I max 1.42 AFigure 1. Generalized block diagram of observer based MPC system Motor maximum torque Tmax 0.068 N.mFrom (1) and (16), the dynamic of the observer error becomes Motor torque constant km 0.0502 N.m/A ~ (t  1)  ( A  LC ) ~ (t )  B~ (t ). x x e  Motor armature resistance Rm 10.6 Ohmwhich together with (16) results in; Motor armature inductance Lm 0.82 mH  ~(t  1)  A  LC x B   ~(t ) x Open loop time constant T 0.0929 s  ~(t  1)    MC   ~ (t ).  Open loop steady state gain K 19.9 rad/(V.s) e   0  e © 2013 ACEEE 26DOI: 01.IJCSI.4.1.1064
  5. 5. Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013Representing (21) in Laplace domain as follows: R m I m  Lm I m s  Vm  k m  m . The armature electrical time constant of system is representedas, Lm e  . R m Expressing the torque equation into the Laplace domain, J eq  m s  k m I m  Td . Combining (21) and (23), Figure 3. Per iteration computation time comparison of QP solving 2 2 methods k m  m ( s ) k m Vm (s ) servomotor. J eq  m ( s )s    Td ( s ).  Rm Rm Substituting the parameter values from Table I in (29), the state space representation of DC servomotor model can beTransfer function from motor shaft position to the motor input obtained as,voltage can be given as,  ( s) 0 1   0  Gm ( s )  .  A  , B   210 , C  1 0, D  0. (30) Vm ( s ) 0  1  The plant system parameters can be expressed as, QET DC servomotor model as given in (30) was considered for the MATLAB simulation of MPC. The simulation results km of the proposed Observer based MPC and high pass filter Gm ( s )  2 . (HPF) based MPC are shown in Fig. 4, Fig. 5, and Fig. 6.  k  Rm  J eq s  m     Rm Simplifying the above equation further, b K G m ( s)   .  s ( s  a) s ( s   ) kmwhere, 2 and b km J eq Rm a J eq Rmkm is the steady state gain and  is the time constant of the Figure 4. Comparative plot of position responsesystem.Converting the transfer function into the state space form, From the results it is observed that observer does better state estimation compared to HPF in transient time. Due to 0 1   0  better state estimation of actual states, control input is A 1  , B   K  , C  1 0, D  0. conservative and states have less oscillation in the transient 0          response. VI. MATLAB SIMULATIONA. QP Solver The proposed algorithm is coded in MATLAB. Fig. 3,shows per iteration computation time comparison of proposedmethod with QR factorization and MATLAB’s QP solverquadprog.B. MPC Implimentation The designed MPC with PIO is tested throughexperimentation for position control application of a DC Figure 5. Comparative plot of second state© 2013 ACEEE 27DOI: 01.IJCSI.4.1.1064
  6. 6. Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013 Figure 6. Comparative plot of control input VII. HARDWARE IMPLEMENTATION Efficacy of the proposed approach is demonstratedpractically on Quanser QET DC servomotor plant with 0.01ssample time. Figure 8. Quanser QET DC servomotor setup Figure 9. Comparative plot of position response using PID controller and MPC Figure 7. Real-time MPC Implementation strategy for position control of QET DC servomotor The experimental setup has DC servomotor with Q2 USBbased data acquisition card and a PC equipped withproprietary QuaRc 2.1 software. QuaRc provides hardware-in-loop simulation environment [21]. Motor shaft positionθ(t) is measured by an encoder. To provide state estimationof plant, PIO is developed. The proposed real-time MPCimplementation strategy and the experimental setup are shownin Fig. 7 and Fig. 8 respectively. The performance of designed observer based MPCscheme is compared with classical controller i.e. PIDcontroller. Experimentation has been carried out with constantload to DC servomotor as an external disturbance. Fig. 9,shows the plot of position response using PID controller Figure 10. Comparative plot of control input of PID controller and MPCand MPC. It shows that proposed MPC scheme has bettertransient response compare to classical controller. Fig. 11, shows the position responses of the plant using Plot of control input of MPC and PID controller is shown two different prediction horizon with the control horizon keptin Fig.10. Control input of MPC is conservative as compare constant at N u  3 . From Fig. 12, it is observed thatto PID, which can be clearly observed from the plot.© 2013 ACEEE 28DOI: 01.IJCSI.4.1.1064
  7. 7. Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013 ACKNOWLEDGMENT The authors would like to thank Rashmi More from De- partment of Mathematics, College of Engineering, Pune, for her sustained contribution towards this work. REFERENCES [1] A. G. Wills, W. P. Heath, “Interior-Point Methods for Linear Model Predictive Control,” Control 2004, University of Bath, UK, 2004. [2] A. G. Wills, A. Mills, B. Ninness, “FPGA Implementation of an Interior-Point Solution for Linear Model Predictive Control,” Preprints of the 18 th IFAC World Congress Milano (Italy), pp. 14527-14532, 2011. Figure 11. Plant response to reference position for different [3] C. V. Rao, S. J. Wright, J. B. Rawlings, “Application of interior prediction horizon point methods to model predictive control,” Journal of Optimization Theory and Applications, pp.723-757, 1998. [4] K. V. Ling, S. P. Yue, and J. M. Maciejowski, “A FPGA Implementation of Model Predictive Control,” Proceedings of the 2006 American Control Conference, pp. 1930-1935. Minneapolis, Minnesota, USA, 2011. [5] A. Sudarsanam, T. Hauser, A. Dasu, S. Young, “A Power Efficient Linear Equation Solver on a Multi-FPGA Accelerator,” International Journal of Computers and Applications, Vol. 32(1), pp. 1-19, 2010. [6] W. Chai, D. Jiao, “An LU Decomposition Based Direct Integral Equation Solver of Linear Complexity and Higher-Order Accuracy for Large-Scale Interconnect Extraction,” IEEE Transactions on Advanced Packaging, Vol. 33(4), pp. 794- 803, 2010. [7] S. Belale, B. Shafai, “Robust Control System Design with a Figure 12. Control input for different prediction horizon Proportional Integral Observer,” International Journal of Control, Vol. 50(1), pp. 97-111, 1989.N p  10 results in a shorter rise time and a shorter settling [8] S. Belale, B. Shafai, H. H. Niemann, J. L. Stoustrup, “LTR Design of Discrete-Time Proportional-Integral Observers,”time in the position response then N p  40 . Thus the IEEE Transactions on Automatic Control, Vol. 41(7), pp. 1056- 1062, 1996.responses are faster while decreasing the Prediction [9] B. Shafai, H. M. Oloomi, “Output Derivative Estimation and Disturbance Attenuation using PI Observer with ApplicationHorizon N p . Fig. 12, shows that this improvement is achieved to a Class of Nonlinear Systems,” IEEE Conferenceat the cost of larger control signal. proceedings, pp. 1056-1062, 2003. [10] S. Belale, B. Shafai, H. H. Niemann, J. L. Stoustrup, “LTR CONCLUSIONS Design of Discrete-Time Proportional-Integral Observers,” IEEE Transactions on Power Electronics, Vol. 15(6), pp. 1056- In this paper, LU factorization based linear solver for 1028, 2000.solution of IPM is presented, which is used for real-time [11] W. H. Chen, D. Ballance, P. Gawthrop, & J. O’Reilly, “Aimplementation of constrained linear model predictive nonlinear disturbance observer for robotic manipulators,” IEEEcontroller for position control of DC servomotor. The issue Transactions on Industrial Electronics, 47(4), 932–938, 2000.of unavailability of internal state vector for implementation [12] K. S. Low, and H. Zhuang, “Roboust Model Predictive Controlof MPC is addressed by designing a PIO. Simulation results and Observer for Direct Drive Applications,” IEEE Transactions on Power Electronics, Vol. 15, NO. 6, Novemberare presented to demonstrate the effectiveness of PIO. 2000.Simulation result shows the superior performance of observer [13] S. Mehrotra, “On Implementation of a primal-dual interior-to estimate the state vector as well uncertainty compared to point method,” SIAM Journal on Optimization, Vol. 2(4),HPF. From experimentation results it is observed that, the pp.575-601, 1992.proposed scheme solves QP problem efficiently within the [14] J. M. Maciejowski, “Predictive Control with Constraints,”specified sample period and performs well for tracking a Pearson Education Limited, 2002.reference position of DC servomotor which is relatively a [15] S. J. Wright, “Applying new optimization algorithms to modelfast dynamic system. Performance of the designed MPC has predictive control,” Chemical Process Control-V, CACHE,been compared with PID control via experimental results. It is AIChE Symposium Series, Vol. 93(316), pp-147-155, 1997. [16] A. G. Wills, “Technical Report EE04025 - Notes on Linearalso observed that, the controller is robust to handle parameter Model Predictive Control”, 2004.uncertainties and load disturbances.© 2013 ACEEE 29DOI: 01.IJCSI.4.1.1064
  8. 8. Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013[17] T. R. Kruth, “Interior-Point Algorithms for Quadratic [19] A. G. Wills, W. P. Heath, “Interior-Point Methods for Linear Programming,” IMM-M.Sc-2008-19, Technical University of Model Predictive Control,” Technical Report, University of Denmark, 2008. Newcastle, Australia, 2003.[18] I. M. Nejdawi, “An Efficient interior Point Method for [20] F. Stinga, M. Roman, A. Soimu, E. Bobasu, “Optimal and Sequential Quadratic Programming Based Optimal Power MPC Control of the Quanser Flexible Link Experiment,”4th Flow,” IEEE Transactions on Power Systems, Vol. 15(4), pp. WSEAS/IASME International Conference on Dynamical 1179—1183, 2000. Systems and Control (Control’08), pp. 175—180, 2008. [21] “QET QuaRc Integration-instructor manual”, Quanser Inc., Markham, ON, Canada, 2010.© 2013 ACEEE 30DOI: 01.IJCSI.4.1.1064

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