Optimization in Predictive Control Algorithm

Transcription

1 Latest rends in Circits, Systems, Sinal Processin and Atomatic Control Optimization in Predictive Control Alorithm JAN ANOŠ, MAREK KUBALČÍK omas Bata University in Zlín, Faclty of Applied Informatics Nám.. G. Masaryka 5555, 76 1 Zlín CZECH REPUBLIC antos@fai.tb.cz, kbalcik@fai.tb.cz he term predictive control desinates a class of control methods which are sitable for control of varios kinds of systems. Predictive control is essentially based on discrete or sampled models of a process. It is possible to se varios models which describe a real system sfficiently accrately. he model is sed for comptation of predictions of the systems otpt. On the basis of the predictions and past vales of the inpt and otpt of the system we can optimize the control process. A cost fnction which is optimized is a fnction of difference of the maniplated variable. An analytical soltion can be obtained by differentiation of the cost fnction. One of the major advantaes of predictive control is its ability to do on-line constraints handlin in a systematic way. For a constrained case the analytical soltion may not be located in an allowed area and we need to se another way of optimization. he optimization problem is necessary to be solved in each samplin period, which is comptationally demandin. Varios kinds of optimization alorithm can be sed. he contribtion is focsed on an analysis of the cost fnction and effects of constraints to an allowed area. he analysis can help with choosin of sitable methods and alorithms. Key-Words: - predictive control, cost fnction, constraints, optimization, simlation 1 Introdction he essential idea of predictive control [1], [], [3], [4] is based on the possibility to predict behavior of a system sin its model. Predictive control can be divided into several parts. A predictor defines relation between past and ftre vales. here are a lot of methods how to obtain prediction eqations. hese methods are based on a model of the controlled system. A rane of varios models can be sed (for instance transfer fnction, ARMA, neral network etc.). A widely sed model in predictive control is the CARIMA model [5] which directly contains a difference of the maniplated variable. his model can be written in the followin form C Ayk =B k 1 + nk (1) where polynomials A and B describes a transfer fnction of the system. Δ = 1 z -1, y is the otpt variable, is the maniplated vale and n is a nonmeasrable noise which is assmed to have zero mean vale and constant covariance. C is a colorin polynomial. On the basis of this model the predictor can be calclated. he predictor [6] can be divided into two parts: a free response, which is that part of the systems response which is detered by past vales of the systems inpts and otpts, and a forced response, which is detered by ftre control increments. y y = GΔ + X () where y and are ftre vales, and are past vales. Matrices G and X contain coefficients which are necessary to be calclated in order to obtain the predictor. he predictor can be also written in the form shown in eq (3) y ˆ = GΔ + (3) y where y is the free response of the process. Matrix G contains vales of the step seqence and it can be written in the followin form 1 G (4) 1 N 1 N N 3 Another part of predictive control is a cost fnction which can be defined as a sm of sqares of control errors and sqares of differences of the ISBN:

2 Latest rends in Circits, Systems, Sinal Processin and Atomatic Control maniplated variable. he cost fnction can be written accordin to eq. (5) as J = yˆ - w yˆ - w+ λ J = GΔ + y - w GΔ + y - w + λ (5) where λ is a weihtin factor which is another deree of freedom. It can be also written in the form, as presented in eq. (6) J c H (6) where is a radient of the cost fnction and H is the Hessian matrix. he radient and Hessian matrix can be written in the form shown in eq. (7) G y w (7) H G G λi he soltion can simply be obtained by derivative of the cost fnction. Eqation (8) considers the vector of the maniplated variable 1 H K w (8) y 1 K G G λi G. where Only the first element is sed and the whole procedre is repeated in the next samplin period. It is called the Recedin Horizon concept. here are three important horizons in predictive control. N 1 and N are imm and imm prediction horizons. It means horizons over which are predicted otpt vales. N is called a control horizon and it defines lenth of the vector of control increments. In technical practice often occr constraints of variables which cases that the analytical soltion of the cost fnction is located otside the allowed area. In a constrained case alternative methods of optimization mst be sed to obtain the soltion. he optimization mst be as effective as possible. In the past, the predictive control was mostly applied for control of systems with lare time constants. Nowadays it is also possible to apply it for control of systems with faster dynamics becase of increasin comptational power. Bt the optimization problem mst be solved in each samplin period and comptationally effective alorithms are reqired. he aim of the paper is to analyze the cost fnction and effects of constraints to an allowed area. Constraints and cost fnction Comptational costs of solvin the optimization problem is dependent on a shape of the cost fnction and the allowed area..1 Cost fnction As it is presented in eq. (5) the cost fnction is a qadratic fnction with the shape depicted in Fi. 1. Fi. 1. Cost fnction Dimension of the cost fnction is eqal to N + 1. It is not possible to show the n-dimensional fnction in 3D space in case N >. Bt it is possible to depict its cts with fixed vales in other axes. In Fi. 1 is depicted the cost fnction for the first samplin step of the control process. In the followin steps the cost fnction looks similarly. Accordin to its shape it can be solved by the derivative. he cost fnction is nimodal with one local imm..1 Constraints he vales of variables in a real system are sally constrained. he constraints are mostly iven by secrity conditions or technical limits. In this case the cost fnction is limited by a specific area. We can consider three types of constraints. he first one is constraint of difference of the maniplated variable. It can be written in the form represented in eq. (9). Δ Δ Δ (9) It can be also expressed by a set of eqations (1) k k 1 k k 1 (1) ISBN:

3 Latest rends in Circits, Systems, Sinal Processin and Atomatic Control he system of eqations can be written in a matrix form, as presented in eq. (11) I 1Δ Δ I 1Δ (11) AΔ b where I is an identity matrix of dimension N N. Constraints of difference of the maniplated variable leads to the shape of the allowed area in the form of nd-cbe which is depicted in Fi.. 3. he shape of the allowed area is depicted in Fi. Fi.. Constraints of difference of maniplated variable he red point is a imm obtained by derivative of the cost fnction. he reen point is the lobal imm of the allowed area. he next type is the constraint of the maniplated variable. It can be expressed by eqation (1). (1) A set of eqations is in form (13). k k 1 k k 1 k 1 k k 1 k k 1 k 1 And the matrix form can be expressed as follows 1 1k 1 A, b 1 1k 1 (14) where is a lower trianlar matrix with ones in the non-zero positions. (13) Fi. 3. Constraint of maniplated variable As it can be seen in fire 3 the rane of k is in the interval form. Bt the next difference vale k 1 is dependent on the previos vale of k. he interval of the vertical axis is bein continosly moved which cases the shape depicted in Fi. 3. Every next interval of other axes in case of a mltidimensional problem is also moved accordin to previos vales of differences of the maniplated variable. he allowed area is formed as a sbspace in the nd space. he third type of constraint is defined as an interval of the otpt variable which can be written in the followin form y y y (15) here mst be defined a relation between y and k. Eqation (3) can be sbstitted to eqation (15) and the constraint is expressed as follows y GΔ + y y (16) Eqation (16) can be modified GΔ y y G (17) Δ y y he matrices A and b take the followin form G 1y y A, b (18) G 1y y If the mappin between maniplated and otpt vales is feasible then the interval of otpt constraint corresponds to the interval of the maniplated variable constraint. In this case the ISBN:

4 Latest rends in Circits, Systems, Sinal Processin and Atomatic Control shape of the allowed area, which is depicted in Fi. 4, is similar to the shape depicted in Fi. 3. Fi. 4. Constraint of otpt variable It is obvios that positions of the local and lobal imms can be different and they are dependent on the cost fnction and constrains. he constraints can be combined each other and the allowed area is in the intersection of them, which is depicted in Fi Simlation and alorithms.1 Simlation As a simlation example is presented control of the followin system of the second order G s (19) 519.9s 1 4.8s 1 he horizons were set as N 1 = 1, N = 3 and N = in order to have a possibility to show raphs (3D cost fnction and D allowed area). he constraints were set to the vales shown in eq. () () y 31 he optimization problem was solved by Hill Climbin alorithm [7]. ime responses of control are depicted in Fi. 6 and Fi. 7. w t, y t C Fi. 6. Setpoint and otpt variables of control process t s 1 8 t V 6 4 Fi. 5. Combination of constraints he allowed area is continos bt we can consider other types of constraints e.. discrete vales, mlti-intervals etc. In these cases a soltion of the optimization problem may be more complicated becase of the shape of the allowed area Fi. 7. Maniplated variable of control process t s We can stdy the shape of the allowed area and vales of the maniplated variables. It is obvios that the control error for k = 1 is lare and there are also defined constraints of difference of the maniplated variable, maniplated variable and otpt variable. he shape of the allowed area is depicted in Fi. 8. ISBN:

5 Latest rends in Circits, Systems, Sinal Processin and Atomatic Control Fi. 8. Allowed area for k = 1 It is obvios that the analytical soltion is located otside the allowed area. For the nconstrained case the analytically compted vale is k Bt the soltion which was fond within the allowed area is 3.5 becase constraint of difference of the maniplated variable is he allowed area for k = is depicted in Fi. 9. Fi. 1. Allowed area for k = 3 In the third step the analytical soltion is inside the allowed area. In most cases the analytical soltion is inside the allowed area. In these cases the soltion can be obtained by derivative of the cost fnction. In other cases mst be sed a sitable alorithm in order to find a soltion inside the allowed area. hese cases are mainly located arond sdden chanes of the reference sinal..1 Alorithms he searchin of the soltion can be divided into two phases. In the first stae is bein searched an arbitrary soltion which is placed inside the allowed area. he second phase consists of searchin the best soltion within the allowed area. Fi. 9. Allowed area for k = In this step the analytical soltion is mch closer to the soltion obtained inside the allowed area. he allowed area for k = 3 is depicted in Fi. 1. Fi. 11. Searchin soltion inside the allowed area he simplest way how to find the allowed area is the random walks method [7]. his method is not sitable for solvin of hiher dimensional problems ISBN:

6 Latest rends in Circits, Systems, Sinal Processin and Atomatic Control becase its efficiency rapidly decreases with increasin dimension. he allowed area can be also fond by the method of penalisation of the cost fnction otside of the allowed area (soft-constraints 1 ). he qestion is how to penalise the cost fnction in order to find the allowed area qickly. For example mltiplication of the cost fnction only increases vale of the cost fnction bt the radient may stay similar. In this case the radient methods may not convere to the allowed area. he analytical soltion can be sed as the initial estimate. Also point [,] is a ood candidate to be inside the allowed area. Bt zero vales of difference of the maniplated variable correspond to the constant vale of the maniplated variable and the free response of the system may not flfil reqirements on constraints of the otpt variable. he next method how to find the allowed area is to solve a system of ineqalities with fixed ftre vales of difference of maniplated variables. he second phase can be solved by radient methods becase of the shape of the cost fnction. A eneral eqation of the radient method can be written in the form shown in eq. (1) xi 1 xi f x (1) where f x is a fnction of the radient of the cost fnction. One of the most sitable methods is qadratic proram [8], [9]. his method is often sed in predictive control bt it is also possible to se other methods which enable to solve this problem sch as evoltionary alorithms [7]. Disadvantae of qadratic proram is hiher comptational cost and comptational time. In cases when a model, initial conditions and a setpoint are known the problem can be solved offline [1]. 4 Conclsion In the paper were introdced basic aspects of solvin the optimization problem in predictive control. In an nconstrained case it is possible to find the soltion by derivate of the cost fnction. he cost fnction is a qadratic fnction, which is nimodal, and it has one local imm. In a 1 In case of soft-constraints cost fnctions of points which are located otside of the allowed area are penalized. constrained case a shape of the allowed area depends on a type and vales of constraints. In this case the analytical soltion may be located otside of the allowed area. here are lots of alorithms which enable to solve this problem. One of the most effective methods is called qadratic prora bt it is also possible to solve this problem by many other methods for example by methods based on evoltionary alorithms. he crcial isse of the optimization are comptational costs becase it mst be solved on-line in each samplin period. Searchin of the soltion can be divided into two phases. In the first phase there mst be fond the allowed area. In the second phase there mst be fond the best soltion inside the allowed area. Acknowledments: Athors are thankfl to Internal Grant Aency (IGA/FAI/14/9) of omas Bata University in Zlín, Czech Repblic for financial spport. References: [1] D. W. Clarke, C. Mohtadi, P. S. ffs, Generalized predictive control, part I: the basic alorithm. Atomatica, 3, 1987, [] D. W. Clarke, C. Mohtadi, P. S. ffs, Generalized predictive control, part II: extensions and interpretations. Atomatica, 3, 1987, [3] M. Morari, J. H. Lee, Model predictive control: past, present and ftre. Compters and Chemical Enineerin, 3, 1999, [4] R. R. Bitmead, M. Gevers, V. Hertz, Adaptive Optimal Control. he hinkin Man s GPC, Prentice Hall, Enlewood Cliffs, New Jersey, 199. [5] E. F. Camacho, C. Bordons, Model Predictive Control, Spriner-Verla, London, 4. [6] J. A. Rossiter, Model Based Predictive Control: a Practical Approach, CRC Press, 3. [7] Back., Foel D. B., Michalewicz Z., Handbook of evoltionary alorithms, Oxford University Press, 1997, ISBN [8] G.M. Lee, N.N. am, and N.D. Yen, Qadratic Proram and Affine Variational Ineqalities: A Qalitative Stdy, Spriner, 5. [9] Z. Dostál, Optimal Qadratic Proram Alorithms: With Applications to Variational Ineqalities, New York: Spriner, 9. [1] Kvasnica, M.; Raova, I.; Fikar, M., "Atomatic code eneration for real-time implementation of Model Predictive Control," Compter-Aided Control System Desin (CACSD), 1 IEEE International Symposim on, vol., no., pp.993,998, 8-1 Sept. 1 ISBN: