On Fractional Predictive PID Controller Design Method Emmanuel Edet*. Reza Katebi.** * echnology and Innovation Centre, Level 4, Deartment of Electronic and Electrical Engineering, University of Strathclyde, Glasgow,G RD. UK.( e-mail: emmanuel.edet@strath.ac.u. ** Industrial Control Centre, Royal College Building, Electronic and Electrical Engineering Deartment, University of Strathclyde,4 George Street,Glasgow,G XW.UK.(el: +444548497; e-mail:m.r.atebi@strath.ac.u Abstract: A new method of designing fractional-order redictive PID controller with similar features to model based redictive controllers (MPC is considered. A general state sace model of lant is assumed to be available and the model is augmented for rediction of future oututs. hereafter, a structured cost function is defined which retains the design obective of fractional-order redictive PI controller. he resultant controller retains inherent benefits of model-based redictive control but with better erformance. Simulations results are resented to show imroved benefits of the roosed design method over dynamic matrix control (DMC algorithm. One maor contribution is that the new controller structure, which is a fractional-order redictive PI controller, retains combined benefits of conventional redictive control algorithm and robust features of fractional-order PID controller. Keywords: Fractional-order PI, Dynamic matrix control, Model-based redictive controller. INRODUCION he idea of incororating future set-oint into control formulation has gained oularity in various forms giving rise to different variants of model-based redictive control (MPC methods. hese MPC algorithms include: generalised redictive control (GPC, dynamic matrix control (DMC, model algorithmic control (MAC or finite sectrum assignment (FSA. he central feature is the same which is the incororation of future set oint in the control law (Camacho & Bourdons, 998. However, each algorithm differ slightly in the structure of the obective function being otimised and in the tye of model used for rediction of future set oint. Constraints handling and inherent multi-inut multi-outut (MIMO caability are two attractive benefits of using redictive control method. he goal of the design rocedure is to realise an otimal or sub-otimal gains of controller that guarantees excellent tracing erformance (Wang, 9. In this aer, attractive features of both MPC and fractional-order PID (FOPID controllers are combined to realise a hybrid controller design algorithm with imroved erformance. A single inut single outut (SISO rocess is selected to test the erformance of roosed controller and results can be directly extended to multivariable control roblems in a straightforward way by modifying dimensions of relevant matrices. Historically, fractional order control systems have been of interest to researchers for many years. his is because these controllers, when roerly tuned, have been found to yield better erformance comared to integer-order controllers under similar conditions (Li & Chen, 4. Several authors have recorded tremendous rogress in utilising fractional order calculus to design control systems (Mone, et al., 8; Padula & Visioli, ; Zhuang, et al., 8. Imroved erformance is artly down to extra flexibility granted by the non-restriction of derivative orders imlying that more conflicting design obectives can be achieved. For instance, a conventional integrator (with integer order is sufficient for reection of steady state error. However, with the fractional case, extra tuning arameters are available for meeting more frequency domain-based stability secifications. As a result of such comarative benefits, many design methods have been roosed over the years for fractional-order PID controllers, esecially, SISO rocess alications designed in continuous time domain. One class of design methods is based on direct synthesis. FOPID controller arameters can be comuted directly in frequency domain to meet some desired secifications such as hase margin, constant hase at gain cross over frequency and other sensitivity constraints (Luo, et al., ; Luo & Chen, 9; Vale rio & Costa, 6. Other methods of designing FOPID controllers include otimization based methods such as linear matrix inequality (LMI and evolutionary algorithms (Lee & Chang, ; Song, et al.,. Considering discrete FOPID controllers with redictive caability, there is no standardised design or tuning method comared to conventional PID controllers. In this aer, a new fractional order redictive PI (FOPPI controller is roosed with hybrid benefits. Future set-oint is incororated in formulating the control law. It also retains both constraints handling caability of MPC controller and robust features of FOPID controllers. Some useful definitions are resented next.. Fractional Order Derivatives Consider a differential equation with fractional order (α: y( t D u( t u( t wo imortant fractional derivative definitions are stated:
Grunwald-Letniov (GL definition Riemann-Liouville Definition. According to Riemann-Liouville: n d ( D f ( t n dt ( n ( where t > ; n- < α < n; n t f d ( n t (Podlubny, 999. An alternative definition is Grunwald-Letniov (GL: D f ( t t lim f ( where: t > ; n- < α < n; n reresents binomial coefficient. ( In continuous form, Lalace transform has been used directly to describe FOPID controllers as given in (3. I C( s Ds (3 s where: and are the integral and derivative orders., I and D are roortional, integral and derivative gains resectively. However, in discrete time domain, different aroximation methods are sometimes used to aroximate fractional integral and derivative terms. hese aroximation methods are based on samling continuous time signal and discretising it at each samling instant. Examles are ustin aroximation, forward Euler aroximation, bacward Euler aroximation, aylor series aroximation and the long memory discrete time PID - LDPID. Although the GL definition results in infinite series, an aroximated form given in (4 is used extensively by several authors for deriving numerical solution (Zhuang, et al., 8. N( t ( tl t (4 D f ( t b f ( t where: b is binomial coefficient. b ; b b ; b b ; b b. L = memory length; Nt ( min t, L al., 8. (Zhuang, et In this wor, the same aroach is followed to define a discrete fractional-order PI controller which is only used to formulate cost function for the roosed FOPPI controller.. Discrete ime Fractional-order PI controller Consider the fractional order PI controller: u( t e( t D e( t i i t he discrete time form can be exressed as given below in (5. i i s u ( e( b e( (5 where: s = samling time. Other terms are as defined in (4. Guo, et al.,( develoed a combined DMC and FOPID design method with some benefits. Similarly, in this aer, an incremental form of fractional-order PI algorithm is used to formulate obective function but without any derivative term in the cost function. u ( u ( u (. i i i i i s u ( e( b e( (6 u ( u ( e( e( i i is be( be( u ( e( e( i is e( b e( (7 u i ( e( is e( Be( (8 where: B b. Let: KI is Equation (8 is therefore re-written: u i ( e( KI e( Be( (9 wo distinct comonents are evident in (9 namely: a roortional term and a fractional integral comonent. hese two terms are used to formulate a new cost function which is minimised in order to obtain the roosed FOPPI controller. However, u ( is not imlemented directly as given in (9. i Direct imlementation of FOPID controller has to be band limited. Instead, a model-based redictive control framewor is used for imlementation of the fractional order redictive PI controller. Consequently, control signal limits are defined inherently using constraint handling feature of MPC.. HE PROPOSED ALGORIHM. A Brief Review of Model Predictive Control Algorithm Consider a SISO rocess described by the state sace equation in (:
x( Ax( Bu( y( Cx( Du( ( where: u is the control inut; y is the outut; x is the state variable vector and matrices A, B, C, and D are system matrix, inut control matrix, outut matrix and feedthrough matrix. Equation ( is usually used (without the D term for DMC derivation. his is because in receding horizon control, current information of the lant is required for rediction and control imlying that the inut u( cannot directly affect the outut y( at the same time. he state sace model is rewritten: he rediction equation is therefore given by: where: CA CA F CA Yˆ ( F Xˆ( G u( (5 ; CB CAB CB G CA B CAB CB 3 CA B CA B CA B... CA m Bxm he cost function to be minimised is given by J: x( Ax( Bu( y( Cx( ( (6 J r( i y( i Q U ( R i N he standard state sace model in ( is augmented in order to incororate integral action. Incremental model is given as: u( u( u( x( x( x( Ax( B u( y( y( y( Cx( Cx( y( Cx( y( CAx( CBu( he augmented matrix equation obtained is given in (: x A x( B u ( x CA I y( CB x ( y ( [ ] y ( where: x x( ; x y(. ( his augmented state sace model is used for rediction of future states. Let the rediction horizon be reresented as and control horizon be N. Given that N, future states can be redicted as follows: xˆ( Ax( B u( xˆ( A x( AB u( B u( xˆ( A x( A B u( N A Bu( N Similarly, the future outut variables can be redicted using: yˆ( CAx( CB u( yˆ( CA x( CAB u( CB u( yˆ( CA x( CA B u(... N CA Bu( N (3 (4 where: R = weighting vector for the control effort and Q = state weighting matrix. A standard MPC solution exist for this otimisation roblem assuming no inut constraint and taing derivative of J. herefore: mc ( U l G G I G e (7 where: e( reresents error at samling instant l [,,,...,] his is the standard MPC control technique with feedbac achieved through receding horizon control (Miller, et al., 999. In contrast, the cost function given in (6 is changed to obtain a new fractional-order redictive PI controller design method. Both roortional error and integral error comonents are introduced in the new cost function. Imlementation of the derived FOPPI controller is by receding horizon technique as only the first column of control inut is alied to the rocess. At a new samling instant, all rediction matrix coefficients are re-calculated.. Derivation of Proosed Control Law In this section, the roosed control law is derived. At a samling instant, within a rediction horizon, the aim of the control system is to bring the future redicted outut Y ˆ( as close as ossible to the exected set-oint signal - Yd ( assuming the set oint signal remains constant in the otimization window. he tas is therefore reduced to finding the otimum (or best control arameter vector ΔU such that an error function between the set-oint and the redicted outut is minimized. he state sace model earlier augmented is used to redict the outut as exlained in revious section. Given: P = rediction horizon and N = control horizon he structured quadratic cost function to be minimised is formed directly from (9 as shown:
where: Let f ( ( e K P I e N J u( K I B e( Y ( control signal weight. = forced outut signal due to the control inut. (8 he future outut after stes is the sum of free outut resonse and forced outut as shown: Yˆ( Yˆ ( Y ( he error signal is the difference between the desired set-oint and the redicted outut: E( Y ( Yˆ ( Given that the forced outut is described as: d Y ( G u( f E( Y ( Yˆ ( G u( (9 d Re-write the cost function given in (8 as vectors: I I J e e K e e K B e e u u ( Error signals can be exressed using (9: e Y Yˆ Gu; e Y Yˆ G u d d e Y Yˆ G u d where coefficient matrices are defined as given below: g g g G gm gm gm g g g g g m Elements of matrix G are as derived reviously: CB CAB G CA B CAB CB CB f xm 3 CA B CA B CA B... CA m Bxm Matrices (G and G are derived from matrix-g: G g g g g g g g g g g m m xm G [ G, G,..., G ]; =,,3,...,. G g G g g g g g G m xm g g g g g g 3 m xm Substituting for error signals in J as in (9: ˆ J [ Y Y G u] [ Y Yˆ G u] ˆ K [ Y Y Gu] [ Y Yˆ Gu] ˆ [ ] [ ˆ Y Y G u Y Y G u] u u d d I d d d d he minimum oint can be found by finding the gradient of J with resect to. u J ; u ( ( ˆ G Y G Y Y ( K G K G( Y Yˆ d d I I d ( G G( Y Yˆ d I u u( G G u( K G G u ( G G where: KI b KI B Solving for otimum control increment: where : ( ( ˆ u ( ˆ Gl Yd Y KIG l Yd Y ( G l( Y Yˆ d l [,,,...,] GG KIG G ( G G I.
Equation ( describes the new FOPPI controller. PI arameters ( and I are rovided for tuning. 3. HANDLING OF CONSRAINS Constraints can be readily formulated into the cost function thereby turning the control design roblem into a quadratic rogramming roblem (Katebi & Moradi,. Several quadratic rogramming routines are available for solving constrained otimisation roblem. he most common constraint is the constraint on inut control signal amlitude. e.g. u (.8 his tye of constraint can be defined to handle inut saturation roblems. Other tyes of constraints include: Constraint on the incremental variation of the inut control signal (e.g..8 u (.8 Constraint on outut signal (e.g.. y (.. 4. SIMULAION EXAMPLES 4. Examle Double Integrator Process Control Consider a double integrator lant given by: x( Ax( Bu( y( Cx(.5 where: A ; B ; C =..5 Let the samling eriod be.s and rediction horizon chosen to be 7. he number of otimization ste chosen to be 7 and all weighting matrices chosen as unity identity matrices of roer dimension. Small values of and i can be used to tune the lant. Here, i and are.35 and.7x -4 with fractional order set to.7 and otimization is unconstrained. Simulation of ste resonse is illustrated in Fig.. U Control signal Outut 7.5.5.5.5.5.55.68.57..39.39.695.348.74.43.5 x....5.3.. 3 4 5 6 7 8 Samling Instant.4. Setoint signal Proosed FOPPI Outut Control Signal -. 3 4 5 6 7 8 Samling Instant 4. Examle Non-Minimum Phase Control Comarison Consider a non-minimum hase rocess control examle where the model is described using state sace with matrices:.7.34.34 A ; B = ; C =..34.763.364 (Uren & Schoor, he samling eriod s is selected as s. Prediction horizon is chosen to be 7 and fractional order of.9. he number of otimization stes is chosen to be 7 and all weight matrices chosen as unity identity matrices of roer dimension. Obtained control matrix is given: U.5539.75.6584.54.38.49.5.4.77.666.3.83.377.65 x..3.3.... With i and equal to.5, excellent set-oint tracing is achieved as shown in Fig.. In order to demonstrate comarative benefit, DMC is used as a baseline (Uren & Schoor,. It was roerly tuned for this same lant by the authors using N =; N c = 3; s is selected as and weight R =.9I. he resultant (incremental DMC control: U mc.53.479 3 x.9 Alied control signal is udated ( u u u. Fig. 3 shows disturbance reection roerty of these two controllers. A 5% inut disturbance is introduced at s. It can be observed that the control inut of the roosed fractional redictive controller falls within the range: u.8. Outut Control signal.5 Proosed FOPPI Outut.5 Setoint Signal DMC Outut 3 4 5 6 Samling Instant.4..8.6 DMC control signal Proosed FOPPI control signal.4 3 4 5 6 Samling Instant Fig.. Ste resonse diagram showing the roosed outut (brown rising to the unit ste reference with zero overshoot comared to DMC outut (blue. Fig.. Ste resonse diagram showing the outut signal rising to the unit ste reference with control inut in blue.
Outut Control signal.5 Proosed FOPPI Outut.5 Disturbance Signal DMC Outut Setoint signal 3 4 5 6 Samling Instant.5.5 DMC Control signal Proosed FOPPI signal 3 4 5 6 Samling Instant Fig. 3. Disturbance reection comarison Both methods have similar settling time of 5s after 5% disturbance (green is introduced at =5s.he roosed method (brown have zero overshoot. 6. CONCLUSIONS It can be observed from simulation examles that the new fractional redictive control design method erforms better than dynamic matrix control algorithm without any increased comutational overhead. Also, control effort used to achieve this result is smaller therefore meeting any inut constraints more easily. he significance of the roosed design is in simlicity of tuning. While the algorithm can be comuted in some rogrammable or comuterised fashion, the entire design rocedure can be automated to the oint of ushing and I nobs. his is exected to be attractive to industrial ractitioners. REFERENCES Camacho, E. & Bourdons, C., 998. Model Predictive Control. London: Sringer-Verlag. Guo, W., Wen, J. & Zhuo, W.,. Fractional-order PID Dynamic Matrix Control Algorithm based on ime Domain. Jinan, China, IEEE. Katebi, M. & Moradi, M.,. Predictive PID Controllers. IEE Proceeding - Control heory Alications, 48(6,. 478-487. Lee, C.-H. & Chang, F.-K.,. Fractional-order PID controller otimization via imroved electromagnetism-lie algorithm. Exert Systems with Alications, 37(,. 887-8878. Luo, Y. & Chen, Y., 9. Fractional order [roortional derivative] controller for a class of fractional order systems. automatica, Volume 45,. 446-45. Miller, R., Shaha, S., Wooda, R. & Kwob, E., 999. Predictive PID. ISA ransactions, Volume 38,. - 3. Mone, C., Vinagre, B., Vicente, F. & Chen, Y., 8. uning and auto-tuning of fractional order controllers for industry alications. Control engineering ractice, 6(7,. 798-8. Padula, F. & Visioli, A.,. uning rules for otimal PID and fractional order PID controllers. Journal of Process Control, (,. 69-8. Podlubny, I., 999. Fractional-Order Sysmtems and PID Controllers. IEEE ransactions on Automatic Control, 44(,. 8-4. Song, X., Chen, Y., eado, I. & Vinagre, B. M.,. Multivariable fractional order PID controller design via LMI aroach. Milano, Italy, International Federation of Automatic Control. Uren, K. & Schoor, G.,. Predictive PID control of non-minimum hase systems, Potchefstroom: Intech. Vale rio, D. & Costa, J. S. d., 6. uning of fractional PID controllers with Ziegler Nichols ye Rules. Signal Processing, 86(,. 77-784. Wang, L., 9. Model Predictive Control System Design and Imlementation with Matlab. London: Sringer Verlag. Zhuang, D., Yu, F. & Lin, Y., 8. Evaluation of a Vehicle Directional Control with a Fractional Order PD Controller. International Journal of Automotive echnology, 9(6,. 679-685. Li, Z. & Chen, Y., 4. Ideal, Simlified and Inverted Decouling of Fractional order IO Processes. Cae own, IFAC. Luo, Y., Chenc, Y. Q., Wang, C. Y. & Pi, Y. G.,. uning fractional order roortional integral controllers for fractional order system. Journal of Process Control, (7,. 83-83.