Modifications of Optimization Algorithms Applied in Multivariable Predictive Control

Transcription

1 WSAS RANSACIONS on SYSS and CONROL are Kubalci, Vladimir Bobal, omas Barot odifications of Optimization Algorithms Applied in ultivariable Predictive Control ARK KUBALCIK, VLADIIR BOBAL Department of Process Control Faculty of Applied Informatics omas Bata University in Zlin Nad Stranemi 45, 75, Zlin CZCH RPUBLIC OAS BARO Department of athematics with Didactics Faculty of ducation University of Ostrava Frani Srama, 79, Ostrava CZCH RPUBLIC Abstract: - Non-linear optimization, particularly quadratic programg (QP), is a mathematical method which is widely applicable in model predictive control (PC). It is significantly important if constraints of variables are considered in PC and the optimization tas is then computationally demanding. he result of the optimization is a vector of future increments of a manipulated variable. he first element of this vector is applied in the next sampling period of PC in the framewor of a receding horizon strategy. In practical realization of a multivariable PC, the optimization is characterized by higher computational complexity. herefore, reduction of the computational complexity of the optimization methods has been widely researched. Besides the generally used numerical Hildreth s method of QP, a possible suitable modification is based on precomputing operations proposed by Wang, L. his general optimization strategy is further modified. wo modifications, which could be applied separately each, were interconnected in this paper. he first modification was published previously; however, its application can be more efficient in connection with the second proposed approach, which modifies precomputing operations. Decreasing of the computational complexity of the optimization by using of the proposal is discussed and analyzed by measurements of floating point operations and control quality criterions using hypotheses tests paired -test and Wilcoxon test. Key-Words: - odel Predictive Control; ultivariable Control; Optimization; Quadratic Programg; Hildreth's ethod; Constraints. Introduction odel predictive control (PC) []-[] has been widely applied in controlling of industrial processes with respect to its ability to deal with control difficulties such as constrained variables [], timedelay [4], nonlinearity [5] and non-imum phase []. heoretical research has a great impact on the industrial world in this area of predictive control. here are many applications of predictive control in industry [7]-[8]. Research of the predictive control has been significantly related to industrial practice. Predictive control is also one of the most effective approaches for control of multivariable systems (IO) [9]. An advantage of model predictive control is that the multivariable systems can be handled in a straightforward manner. A predictive controller can be divided into two subsystems, a predictor and an optimizer, which are cooperating on a receding horizon strategy []. In the PC, the basic idea is to use a model of a controlled process to predict future outputs of the process [] and a trajectory of future manipulated variables is given by solving an optimization problem incorporating a suitable cost function with constraints []. Only the first element of the obtained control sequence is applied in a framewor of the receding horizon strategy. One of the advantages of the predictive control is its ability to do on-line constraints handling in a systematic way, which frequently appears in industrial applications. A significantly important part of the constrained PC is an optimization tas. A frequently used type of optimization in PC is the quadratic programg [], where constraints are considered. Previous, current and predicted control variables of PC are included in a cost function [4]. In case of constrained multivariable predictive control, many constraints are processed in the optimization problem. herefore, a selection of an appropriate numerical method is a necessary condition for successful achievement of the vector of future increments of the manipulated variables. he Hildreth s method [5] has been widely used for purpose of solving of the quadratic programg -ISSN: Volume, 8

2 WSAS RANSACIONS on SYSS and CONROL are Kubalci, Vladimir Bobal, omas Barot problems in PC. his approach can be categorized as a dual method [5], which manipulates with the Lagrangian multipliers [5]. Its modifications applied in PC have not been widely described in literature. However, a general modification, presented by Wang, L., has been published in [5] and is frequently utilized in PC algorithms []. Reduction of a computational complexity is based on testing of the occurrence of a multidimensional extreme, which is computed in the current sampling period in PC under all constraints. In this paper, an observed modification [8] of the optimization strategy following Wang, L. [5] is further improved by interconnecting together with an efficient approach to Hildreth's method [7], which is based on from algorithmic point of view. As evaluation of all defined constraints is significantly time-demanding in multivariable PC, the proposed modification of the Wang s approach can be advantageous due to significant reduction of numerical iterative operations required by the optimization algorithm. ultivariable odel Predictive Control In the multivariable model predictive control []- [], a system with two inputs and two outputs (IO) will be further considered. he IO processes are frequently encountered multivariable processes in practice [9]. A general transfer matrix [] of a IO system can be expressed as (), where U and Y are vectors of the manipulated variables and the controlled variables. z z z z z () z z z Y U () As an example, a model with polynomials of second degree was chosen in ()-(7). his model proved to be effective for control of several IO laboratory processes [7]-[8], where controllers based on a model with polynomials of the first degree failed. he model has sixteen parameters. he matrices A and B are defined as follows: a z az az a4z A z () a5 z a z a7 z a8 z b z b z b z b4 z B z (7) b5 z b z b7 z b8 z A widely used model in model predictive control is the CARIA (Controller Autoregressive Integrated oving Average) model which we can obtain by adding a disturbance model (8), where n is a non-measurable random disturbance that is assumed to have zero mean value and constant covariance and (9) in case of IO system. z z z z A y B u C Δ n (8) z Δ z (9) z For purposes of simplification, polynomial matrix C will be further supposed to be equal to the identity matrix [9]. he difference equations () of the CARIA model are used for computation of predictions in predictive control. hese equations can be further written into a matrix form ()-(). 4 4 y a y a a y a y a y a a y a y b u b u b u b u 4 () U z u z, u z, z y z, y z Y () It may be assumed that the transfer matrix () can be transcribed to form (4) of the matrix fraction. z z z z z A B B A (4) y a b a7 y a7 a8 y a8 y 5 y a5 a y a y u b u b u b u 5 7 B u B u y A y A y A y 8 () he model can be also written in form (5). z z z z A Y B U (5) a a A a5 a, 7 a a a a 4 A a5 a a7 a () 8 -ISSN: Volume, 8

3 WSAS RANSACIONS on SYSS and CONROL are Kubalci, Vladimir Bobal, omas Barot a a 4 A a a, 8 b b B b5 b, 7 b B b b b 4 8 It was necessary to directly compute three stepsahead predictions by establishing of previous predictions to later predictions. he model order defines that computation of one step-ahead prediction is based on the three past values of the system output: B u B u yˆ A y A y A y B u B u yˆ A y A y A y B u B u yˆ A y A y A y () Computation of the predictions can be divided into recursion of the free response and recursion of the matrix of dynamics. he free response vector can be expressed as: p p q q q q4 q5 q y p q q q q q4 q5 q y p p u q q q q4 q5 q y y p4 p4 u q4 q4 q4 q44 q45 q4 y p5 p5 q5 q5 q5 q54 q55 q5 y p p q q q q4 q5 q y P Q Q Q y y P u Q Q Q y Pu Q y P Q Q Q y y (4) Coefficients of matrices P and Q for further predictions are computed recursively. Based on the three previous predictions it is repeatedly computed the next row of the matrices P and Q in the following way: p7 p7 P4 p8 p A P A P A P (5) 8 q7 q7 Q4 q8 q A Q A Q A Q () 8 q7 q74 Q4 q8 q A Q A Q A Q (7) 84 q75 q7 Q4 q85 q A Q A Q A Q (8) 8 Recursion (9)-() of the matrix is similar. he next element of the first column is repeatedly computed and the remaining columns are shifted. his procedure is performed repeatedly until the prediction horizon is achieved. If the control horizon is lower than the prediction horizon a number of columns in the matrix is reduced. Predictions can be written in a compact matrix form (). g, g,,,,,,, g, g, g 4, g 4,, u,, u,, g g u g g g g u u g 4, g 4, g, g, u g 5, g 5, g, g, u (9) g7 g7 4 g8 g A A A () 8 yˆ j Δu j PΔu Qy j () j N. Implementation of PC In the framewor of the optimization subsystem of PC, the computation of a control law of PC is particularly based on imization of quadratic criterion (). his specific form of the optimization problem is then related to quadratic optimization []-[4]. J N j u j j N u e () j where e(+j) is a vector of predicted control errors, Δu(+j) is a vector of future increments of the manipulated variable, N is a length of the prediction horizon, N u is a length of the control horizon and λ is a weighting factor of control increments. A predictor in a vector form is given by: yˆ u y () where ŷ is a vector of system predictions along the horizon of the length N, Δu is a vector of control -ISSN: Volume, 8

4 WSAS RANSACIONS on SYSS and CONROL are Kubalci, Vladimir Bobal, omas Barot increments, y is the free response vector. is a matrix of the dynamics given by equation (4). N (4) where sub-matrices i have dimension x and contain values of the step sequence. he criterion () of the optimization problem can be written in a general vector form (5). J y w yˆ w u u ˆ (5) where w is a vector of the reference trajectory. he criterion can be modified using the expression (5) to (). J g u u Hu () where the gradient g and the Hess matrix H are defined by following expressions: g y w (7) H (8) In context of the quadratic programg optimization with constraints, general formulation of predictive control is as follows g u u Hu (9) u with respect to matrix inequality in a compact form: γ Δu γ m,n R, γ R m m m, m n n mn () insight to the optimization strategy used in PC. PC is characteristic by a frequent occurrence of a situation, when the quadratic programg problem can be completely substituted by a simple multidimensional extreme problem. he main idea of the modification is based on a pre-computed vector of future increments of the manipulated variables in form of a multidimensional extreme (). If the inequality () is fulfilled, then the whole problem of quadratic programg is eliated and the solution has form (). Δu H b () If the multi-dimensional extreme is achieved, then the computational complexity significantly decreases. Otherwise the quadratic programg problem has to be solved using Hildreth s method, which results in equation (). - Δu - H ( d b ) () Proposal of Interconnection of Convenient Optimization Strategies For purposes of further decreasing of computational complexity of the optimization algorithm, the approach described in the previous section was further improved. he approach presented by Wang. L [5] spends a large amount of the computational time by evaluation of all conditions in (). his approach was published in [8]. It was focused on improving of precomputing operations based on constraints in QP. i, i ; m m m m( i) R,n : mδu ; ; Δu H i i i b; i n he condition () can be effectively modified while maintaining the original advantages of the modification presented by Wang. L. A new form of the conditions is defined by (4). he testing of the conditions is progressively divided into partial operations. (). Optimization Algorithm Widely Used in PC In practical applications of PC, a modification proposed by Wang, L. has been widely implemented [5]. his advantageous approach represents a new -ISSN: Volume, 8

5 WSAS RANSACIONS on SYSS and CONROL are Kubalci, Vladimir Bobal, omas Barot, m : Δuˆ γ Δu arg( J; Δu γ), m : Δuˆ γ Δu H b, m : Δuˆ γ Δu arg( J; Δu γ) δ,n δ, R, γ R ; Δuˆ H b; () ( ) γ γ ( ) n n n In case of the first failure the testing is terated and the rest of the conditions is not evaluated. he multi-dimensional extreme problem is then solved. his enables saving of the computational time. his is significant particularly in case of multivariable PC where the number of operations is large. If all the conditions are fulfilled and the testing is completed, then the quadratic programg problem must be solved. he second approach, focused on improving the Hildreth s method, was published in [7]. his method was primarily based on improving the numerical algorithm of the Hildreth s method. he main principles are bound with including of a new exit condition of the iterative algorithm. his modified exit condition did not significantly affect quality of control, as can be seen in simulation results in [7]. Instead the comparison of equality of last computed results in main numerical cycle, further condition of fulfilling constraints () is being tested [7, p. 78]. In this paper, modification [7] is denoted as the first modification. As the second modification, the methodological extension [8] is entitled. 4 Simulation Results PC with both modifications was compared with the PC without modifications by simulation of constrained multivariable predictive control in ALAB. he comparison was based on a measurement of floating point operations [] of a whole PC algorithm which was applied for simulation control of a simulation controlled system defined by (5)-(). A setting of further parameters of control is defined by (7). -.4z.7z A( z ) -.7z.759z.4z.9z -.9z.4z (5).98z -.97z B ( z ).755z.88z max max.9z.779z.8z.5z () u.7, u.75, u.7 (7) Constraints of the manipulated variables and increments of the manipulated variables were considered which is obvious from definition (7). Setting of constraints has an appropriate form, as can be seen in (8). Where I is an identity matrix [9] and is a unit matrix [9].,,, I,,,, I,,,,, I,,, u u( ) u u u( ) u u u( ) u, γ u u( ) u umax umax max max (8) Floating point operations of particular cases of PC were computed using rules in []. he complexity variable, which is equal to a maximum horizon N, was incrementally increasing. As can be seen in able, a significantly lower number of floating point operations F* was achieved when using PC with both modifications. In F, a maximum number of operations was achieved for case of non-modified PC. Application of only the first modification of PC was active in small horizons µ= and µ=5 with a number of operations F. In other cases with horizons greater than µ=5, the second modification got better results, as can be seen in F. Interconnection of both modifications caused decreasing of floating point operations. Simulation results of PC without modifications can be seen in Fig. - and with proposed modifications in Fig ISSN: Volume, 8

6 WSAS RANSACIONS on SYSS and CONROL are Kubalci, Vladimir Bobal, omas Barot w (), y (), u () w() y() u() Fig. Simulation of PC - st Variables of Control without odifications w (), y (), u () w() y() u() Fig. Simulation of PC - nd Variables of Control without odifications w (), y (), u () w() y() u() Fig. Simulation of PC - st Variables of Control with Proposed odifications w (), y (), u () w() y() u() Fig. 4 Simulation of PC - nd Variables of Control with Proposed odifications able : Analysis of Number of Floating Point Operations using Proposed odifications in PC PC without odification PC with st odification PC with nd odification PC with odifications µ F F F F* An order of the computational complexity can be expressed by using a function O=O() []. F * is the number of flops and O * expresses the order of the complexity function for the proposed approach with both modifications. O is a complexity function for the case of PC without modifications. In equations (9) and (4), the results are obtained using a nonlinear regression []. F (9) O O( ) 859 F * 7 48 (4) * O O * ( ) 7 Results of analyses of floating point operations for PC without modifications F and for PC with both modifications F* can be seen in Fig.5 and Fig.. F [flops],+9,+9,+9,+9 5,+8,+ F = 859µ - 47µ +.µ Fig. 5 Analysis of Floating Point Operations for PC without odifications µ -ISSN: Volume, 8

7 WSAS RANSACIONS on SYSS and CONROL are Kubalci, Vladimir Bobal, omas Barot F*[flops],+9,+9,+9,+9 5,+8,+ F* = 7µ + 48µ - +µ µ Fig. Analysis of Floating Point Operations for PC with odifications Further, quality of control was analyzed using criterions (4) and (4) generally used in research focused on control engeneering e.g. [4], [7]. u ( ) u ( ) w ( ) y( ) w ( ) y ( ) J (4) J (4) hese criterions of quality (4)-(4) were measured in whole PC control in case with both proposed modification and without modifications depending on horizons defined by µ, as can be seen in able and able. Where both criterions J and J are depended on a number of horizons µ. able : Analysis of Control Quality using Criterion J =J (µ) PC without odification PC with st odification PC with nd odification PC with odifications µ J J J J 58,9 58,9 58,9 58,9 5 5, 5, 5, 5, 55,97 55,97 55,97 55, ,87 55,8 55,87 55,8 58,99 58,97 58,99 58, ,74 58,74 58,74 58, ,8 58,8 58,8 58, ,495 58,495 58,495 58,495 able : Analysis of Control Quality using Criterion J =J (µ) PC without odification PC with st odification PC with nd odification PC with odifications µ J J J J 95,44 95,44 95,44 95,44 5 9,84 9,84 9,84 9,84 97, 97, 97, 97, 5 9,7 9,7 9,7 9,7 9,959 9,9 9,959 9,9 5 9,9479 9,9479 9,9479 9, ,7495 9,7495 9,7495 9, ,44 9,44 9,44 9,44 According to procedures for testing hypotheses, described e.g. in []-[], the measured criterions in able and able were further tested. At first, testing the normality of data [] was provided using a generally applied method Shapiro- Wil described in [4]. able 4 contains results of testing normality of data. esting zero-hypotheses, which express fulfilling of the normality, were confirmed on the significance level in software PAS [5]. able 4: esting Normality of Data of Criterion J =J (µ) using Shapiro-Wil est Result of esting Zero- Hypothesis ( / ) =,5 =, PC PC PC PC with without with st with nd odif. odif. odif. odif. p-value p-value p-value p-value,978,9785,978,9785 =, able 5: esting Normality of Data of Criterion J =J (µ) using Shapiro-Wil est PC without odif. PC with st odif. PC with nd odif. PC with odif. Result of esting Zero- Hypothesis ( / ) =,5 =, =, p-value p-value p-value p-value,45,45,45,45 -ISSN: Volume, 8

8 WSAS RANSACIONS on SYSS and CONROL are Kubalci, Vladimir Bobal, omas Barot xistence of statically significant differences between J and J with and without modification was tested by paired -test or Wilcoxon test. he paired -test can be applied for data comparison, which indicate normality behaviour. he fulfilling of normality is obvious from able 4 or able 5. In other cases, the Wilcoxon test has to be used for non parametrical data. Depending on testing normality of all data in able 4 and able 5, zero hypothesis about nonexistence of statistically significant differences were performed, as can be seen in able and able 7. Hypotheses, which were failed to reject, express situation with similarities of both data sets (before and after modifications in PC). In results of testing hypotheses, similarities in criterions J and J were found. herefore modifications of PC, proposed in this paper, have not statistically significant influence on the control quality. hese conclusions can be verified using able and able, where differences on the 4th decimal places can be seen. able 5: esting Hypotheses on Non-xistence of Differences between Data of Criterion J =J (µ) before and after Applied odification in PC Result of esting Zero- Hypothesis ( / ) PC without odif. p-value PC with odif. Applied est =,5 p=,: Paired - test =, p=,: Paired - test =, p=,88: Wilcoxon able : esting Hypotheses on Non-xistence of Differences between Data of Criterion J =J (µ) before and after Applied odification in PC Result of esting Zero- Hypothesis ( / ) =,5 =, =, PC without odif. p-value PC with odif. p=,79: p=,79: p=,79: Applied est Paired -test Paired -test Paired -test 7 Conclusion Interconnection of modifications of optimization numerical methods was applied in constrained multivariable PC in this paper. Advantages of the proposed approach were demonstrated and proved by simulations in ALAB. ultivariability and considered constraints in PC significantly increase a computational complexity of the optimization. herefore, the proposed approach can be advantageous for multivariable PC with constraints. Analysis of saving of floating point operations and influence of the proposed approach on quality of control was performed. By analysis of the simulation results it was proved that application of the proposed modification significantly saves floating point operations and concurrently does not affect quality of control. It was then proved that the proposed method can be successfully applied. References: [] J.P. Corriou, Process control: theory and applications, Springer, 4. [] W.H. Kwon, Receding horizon control: model predictive control for state models, Springer, 5. [].F. Camacho, C. Bordons, odel predictive control, Springer, 7. [4]. Kubalci, V. Bobal, Predictive control of multivariable time-delay systems, AC Web Conf., Vol.5, 7. [5] J. Nová, P. Chalupa, Nonlinear State stimation and Predictive Control of ph Neutralization Process, Nostradamus : Prediction, odeling and Analysis of Complex Systems,. []. Barot,. Kubalci, Predictive Control of Non-inimum Phase Systems, 5th International Carpathian Control Conference (ICCC), 4. [7] D. Ingole, J. Holaza, B. aacs,. Kvasnica, FPA-Based xplicit odel Predictive Control for Closed Loop Control of Intravenous Anesthesia, th International Conference on Process Control (PC), 5. [8] D. Honc, F. Duse, State-Space Constrained odel Predictive Control, 7th uropean Conference on odelling and Simulation,. Control of Intravenous Anesthesia, th International Conference on Process Control (PC), 5. [9] P. Navratil, L. Pear, Possible approach to control of multi-variable control loop by using tools for detering optimal control pairs, -ISSN: Volume, 8

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