Possible Way of Control of Multi-variable Control Loop by Using RGA Method

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1 Possible Way of Control of Multi-variable Control Loop by Using GA Method PAVEL NAVATIL, LIBO PEKA Department of Automation and Control Tomas Bata University in Zlin nam. T.G. Masarya 5555, 76 Zlin CZECH EPUBLIC Abstract: - The paper describes one of possible approaches to control of multi-variable control loops. In the approach to control is used the so called GA (elative Gain Array) method, simple approach to ensure of invariance of control loop and simple approach to a design of the so called primary controllers. The GA serve to determine the optimal input-output variable pairings especially for a multi-variable controlled plant. The so called correction members are then generally considered for ensuring invariance of a control loop. Further, it is considered that primary controllers are determined by arbitrary single-variable synthesis method for optimal input-output variable pairings. imulation verifications of the mentioned way of control are carried out for twovariable control loop. Key-Words: - MIMO control loop, GA, simulation, synthesis Introduction Controlled plants with only one output variable (controlled variable) which are controlled by a one input variable (disturbance variable, manipulated variable) are called as IO (ingle-input ingle- Output, single-variable) controlled plants. However, there are not a little cases where it is more than one output variable controlled simultaneously by means of more than one input variable, e.g. aircraft autopilots, heat exchangers, chemical reactors, distillation columns, helicopter, tan processes, steam boilers, steam turbines, etc. [], []. In these cases, it means that there is larger numbers of dependent IO control loops. These control loops are complex and have multiple dependencies and multiple interactions between different input variables (manipulated variables and disturbance variable and output variables (controlled variable. Mentioned control loops are nown as MIMO (Multi-Input Multi-Output, multi-variable) control loop and represent a complex of mutually influencing singlevariable control loops []. pecial case of the MIMO control loop is IO control loop [3]. Multi-variable control methods have received increased industrial interest [4]. It is often no easy to tell when these control methods are necessary for improved performance in practice and when usage of simpler control structures are sufficient. Therefore, it is useful to now functional limits and structure of the whole control loop, i.e. controlled plant, controller and separate signals in the control loop. [] Control methods of MIMO controlled plants can be verified not only by using simulation tools, but also on the laboratory models. ome multi-variable laboratory models have been described in the literature, e.g. helicopter model [5], [6], tan model [], [7], etc. imulation experiments were performed, for chosen MIMO controlled plant, in MATLAB/ IMULINK software [8]. The MATLAB software serves for programming and technical computing in many areas. The IMULINK software is part of the MATLAB environment and serves to analyzing and simulation of dynamics systems. It is possible to use the MATLAB/IMULINK software for education and also for research [9], []. Analysis and Control Design of MIMO Control Loop The controlled plant generally consists of m input variables and n output variables. It is generally a non-square controlled plant type n m. It means, there are three possible cases, i.e. m = n, m > n, m < n. In the next part of the paper, it is mostly considered that controlled plant have a same number of input variables and output variables, i.e. m = n (square controlled plant type n n). One of possible approach to analysis and control design of MIMO control loop, for a controlled plant in steady state, is using the so called the GA (elative Gain Array) method [], []. The GA is useful for MIMO controlled plants that can be decoupled. The IBN:

2 other approach to analysis and control design of MIMO control loop can be found e.g. in [3], [4].. Description of the control loop It is generally considered a MIMO control loop with measurement of disturbance variables shown in the Fig. []. V( W( E( G ( G ( U ( U ( U( G V ( G ( Y V ( Y ( Y( Fig. - Basic scheme of MIMO control loop with measurement of disturbance variables The description of the figure is following, i.e. matrices G (, G (, G V ( and G ( denote transfer function matrices of a controlled plant, controller, measurable disturbance variables and correction members. ignal Y( denotes the Laplace transform of the vector of controlled variables, U( is the Laplace transform of the vector of manipulated variables, V( is the Laplace transform of the vector of measurable disturbance variables, W( is the Laplace transform of the vector of setpoints and E( is the Laplace transform of the vector of control error (E( = W( - Y(). They are considered transfer function matrices of a controlled plant G ( and measurable disturbance variables G V ( in forms n V Vm G GV () n nn Vn Vnm where i Y, i, U (, V i YV i ; V i,,..., n,..., m, m n Transfer function matrices of controller G ( and correction members G ( are considered in the following forms n m G G () n nn n nm where i U, i, E (, i U i ; V i,,..., n,..., m, m n The relations () and () can be used to build other transfer function matrices that occur in the MIMO control loop with measurement of disturbance variables (see Fig. ), i.e. closed loop transfer function matrix G W/Y ( and disturbance transfer function matrix G V/Y (. G G W/Y V/Y I G G( G G( I G G G G G (3) V (4) These transfer function matrices can be use at control design and also to ensuring invariance of the MIMO control loop.. The relative gain array The GA is a tool to the analysis of the interactions between input variables u and output variables y i especially of a MIMO controlled plant. In other words, the GA is a normalized form of the gain matrix that describes the influence of each input variable on each output variable. Each element in the GA () is defined as the open control loop gain divided by the gain between the same two variables when all other loops are under so called "perfect" control []. Further, it is considered a linear n n controlled plant. n Λ (5) n nn where ( yi i ( y i u i u ) u u ) u y l i,,.., n u,,.., n y, l,.., n l l i ( y ) is the open control loop gain with all other control loop open, ( y ) is the open i u control loop gain with all other control loop closed, i is the relative gains for the corresponding variable pairings, i.e. element of the GA. Transfer function matrix of a controlled plant can be written in the following form Y G U( U( G Y (6) where U( and Y( are n-dimensional vectors of inputs and outputs variables and G ( is an n n transfer function matrix of the controlled plant. From relations (5) and (6), it follows that the GA for the controlled plant G ( can be determined as T T Λ ( G ( s )) G ( G ) G () ( G ()) (7) thus i [ G ()] i [ G ()] i y l IBN:

3 where s = = steady state, operator implies an element by element multiplication (Hadamard or chur product). In the [] is presented the procedure to calculate GA for non-square transfer function matrix by using the pseudoinverse. There are some important properties and rules to understanding and analyzing the GA, i.e. eparate elements i are dimensionless and so independent of units. The sum of all the elements i of the GA (5) across any row, or any column will be equal to n i n i i (8) The relation simplifies calculation of elements i, e.g. in case, only element must be calculate to determine all elements, in 3 3 case, only 4 elements must be calculate to determine all elements, etc. Each row in the GA represents one output variable y i and each column in the GA represents one input variable u. The interpretation of the determined elements i in the GA can be classified as follows i = : This implies that u influences y i without any interaction from the other control loop. i = : This means u has no effect on y i. < i < : This indicates that control pair u - y i is influenced by the other control loops. i > : The positive value of the GA indicates that the control pair y i - u represents dominant control loop. Others control loops have an influence on the control pair in the opposite direction. i < : This means that the control pair y i - u causes instability of the control loop. Control pairs y i - u whose input and output variables have positive GA elements ( i ) and their values are close to one are considered as the optimal control pairs [4]. If the value of i fulfils above mentioned general rule the control of the control loop for the control pair y i - u is possible. For other values, the control can become difficult because the interaction rate is too high. Then, it is possible to determine the parameters of the so called primary controllers via classical IO synthesis methods for the optimal control pairs y i - u gained by using the GA pairing method [5]. The GA pairing method has also some shortcoming, i.e. the GA method ignores process dynamics. If the transfer function has very large delay or constant relative to the others, steady state GA analysis provide an incorrect recommendation. In this case it is then preferable to use the so called the NGA (relative normalized gain array) pairing method..3 Invariance of control loop The invariance of the MIMO control loop can be ensured if the transfer function matrix G V/Y ( (4) is zero. It can be possible if the following relation is valid G G G (9) V In the next part of the paper, it is considered an approach which ensures the so called approximate invariance of control loop. It means that influence of disturbance variables is generally eliminated only partially (e.g. only in steady state). Further it is considered the diagonal elements of the transfer function matrix G V ( (b) and G ( (a) have a dominant influence. The influence of the other elements of the transfer function matrix G V ( and G ( is omitted at design of correction members. o that invariance of the control loop is ensured only for diagonal elements. The influence of separate elements of the transfer function matrix G V ( and G ( can be verified by using the GA tool. In this case it is considered that corresponding number of IO branched control loops with measurement of a disturbance variable is used. Connection of all IO branched control loops is the same and they differ in separate transfer functions and variables (see Fig.). []. v w e Fig. - IO branched control loop with measurement of disturbance variable v Transfer function of correction members is determined by using the following relation i V, i, u i,,n,,,n, i V y () where V, are separate elements of transfer matrix G V (, are separate elements of transfer function matrix G (. In the case that diagonal elements of transfer function matrix G V ( or G ( have not dominant influence then described approach to solution of invariance of the control loop may not ensure the desired behaviour. The above mentioned approach for ensuring invariance of control loop by using IO branched control loops with measurement of disturbance variables can also be used for a reduction of interactions of separate non-dominant control loops, i.e. a reduction of influence of non-dominant elements of the transfer function matrix of IBN:

4 controlled plant G ( in the MIMO control loop. uch the control loop is then called decoupling control loop. In this case, it is considered that separate non-diagonal elements of the transfer function matrix of controlled plant G ( represent measurable disturbance variables..4 Control design of control loop One of the possible approaches to control of MIMO control loops is described in the following part. This approach uses analysis of the interactions between input variables and output variables, i.e. the GA method. It can be generally divided a solution this problem into several parts, i.e. determination of parameters of primary controllers then ensuring invariance of control loop and also ensuring at least partial decoupling control loop, i.e. partial reduction of the influence of non-dominant elements (nonoptimal control pair of the transfer function matrix of controlled plant G ( in the MIMO control loop. The so called primary controllers are designed by any synthesis method of IO control loops. Parameters of primary controllers are determined for optimal control pairs gained by using the GA (5), (7) for the controlled plant G (. Invariance of control loop is ensured by using IO branched control loops with measurement of disturbance variables and by determination of correction members via relation (). These correction members are determined only for diagonal elements of transfer function matrix G V ( and G (. For ensuring decoupling control loop is possible to use the approach described in paragraph.3. In this case, the relation (), derived from (), is used, i.e. Pi i i,,,n,, i () The relation () is used for determination of parameters of the so called auxiliary controllers P i. In this case it is considered that aside from diagonal elements of transfer function matrix G (, i.e. i have non-dominant influence and diagonal elements of transfer function matrix G (, i.e. have dominant influence. Modified version of the MIMO branched control loop with measurement of disturbance variables, where the auxiliary controllers are used, is shown in Fig imulation Verification of Described Approach to Control 3. Description of two-variable controlled plant It is considered two-variable controlled plant, i.e. controlled plant having a two input variables and two output variables. The Laplace transform of a vector of an output variable is given by the following relation Y G U G V () V where Y(, U( and V( is the Laplace transform of the vector of controlled variables (Y( = [Y, Y ] T ), manipulated variables (U( = [U, U ] T ) and measurable disturbance variables (V( = [V, V ] T ). Mentioned equation () can be described in the following form Y U V G G V (3) Y U V Transfer function matrices G ( and G V ( are considered in these forms G s 5s 6 s 5s 6 (4).5.5 s 3s s 3s V V G s 5s 6 s 5s 6 (5) V..6 V V s 3s s 3s tep responses of transfer function matrices G ( and G V ( are shown in the following figures (see Fig. 3, Fig. 4). Amplitude To: Out() To: Out() From: In() tep esponse From: In() Time Fig. 3 - tep response of transfer matrix G ( (4) Amplitude To: Out() To: Out()...4. From: In() tep esponse From: In() Time Fig. 4 - tep response of transfer matrix G V ( (5) 3. Control of two-variable control loop The approach to MIMO control described in the paragraph.4 is used for control of the two-variable control loop. First transfer functions of primary controllers for optimal control pairs (5), (7) are determined then approximate invariance of control loop is ensured by using () and finally decoupling control loop is solved by using (). IBN:

5 To determination of optimal control pairs of the controlled plant (4) was used (7), hence () () () () Λ ( G ()) (6) () () () () then T (7) which means that optimal control pairs are y - u and y - u. These optimal control pairs corresponding transfer functions and (a) for which are designed primary controllers of the transfer function matrix of G (, i.e. in this case transfer functions and (a). Parameters these controllers (PI controller were determined via the method of balance tuning [6]. To use this method it was necessary to modify transfer functions and as follows.5.8s.75.44s e, x e, x.6565s.47s (8) then.6s.383 G s s ( ).993s. 86 (9) s Beside above mentioned methods to determine of parameters of primary controllers is possible to use also other IO synthesis methods, e.g. Ziegler Nichols methods, Cohen-Coon method, the method of desired model, Whiteley method, the pole placement method, etc. [], [7]. Correction members were determined from relation (), where was assumed a dominant influence of diagonal elements of transfer function matrix G V (.33 G ().4 Influence diagonal elements of the G V ( (5) was verified by using (7), i.e. determination of the GA for the G V ( T Λ ( GV()) GV() ( GV ()) () Decoupling control loop was ensured by using of non-dominant elements of transfer function matrix G ( (4). Parameters of the auxiliary controllers P were determinate via relation (), i.e. P.8 G P () P.3333 The scheme of modified two-variable control loop is considered according to Fig. 5. v w e v w e P u P u V V V V Fig. 5 - Modified two-variable branched control loop with measurement of disturbance variables 3.3 imulation verification of control loop The MATLAB/IMULINK software [8] is used for simulation verification of proposed control method of the two-variable control loop (see Fig. 5). imulation courses of the two-variable control loop with utilization of chosen IO synthesis method, which is used at design of parameters of primary controllers, are presented in the following figures (see Fig. 6, Fig. 7). The following parameters were chosen and used at all simulation experiments vector of setpoints (t w, t w ): [, 8] vector of setpoints (w, w ): [.7,.7] vector of disturbances (t v, t v ): [4, ] vector of disturbances (v, v ): [.4,.4] total of simulation (t ): 5 step ():.5 y, u, w, v y, u, w, v.5.5 y - controlled variable u - manipulated variable w - setpoint v - disturbance variable y - controlled variable u - manipulated variable w - setpoint v - disturbance variable Fig. 6 - imulation course of control loop with utilization method of balance tuning without the use of auxiliary controllers P and P y y IBN:

6 y, u, w, v y, u, w, v.5 y - controlled variable u - manipulated variable w - setpoint v - disturbance variable y - controlled variable u - manipulated variable w - setpoint v - disturbance variable Fig. 7 - imulation course of control loop with utilization method of balance tuning with the use of auxiliary controllers P and P It is obvious from the simulation courses of control loop shown in the Fig. 6, Fig. 7 and from other simulation experiments that the proposed control method can be used to control of a MIMO control loop. Using this control method it was possible to ensure approximate invariance of control loop (see Fig. 6, Fig. 7) via correction members i () and also decoupling control loop (see Fig. 7 compare to Fig. 6) via auxiliary controllers P i (). 4 Conclusion The goal of the paper was to describe one of the possible approaches to control of MIMO control loops. The control method uses the GA tool for ensuring optimal control pairs for which they are determined parameters of the primary controllers via classical IO synthesis methods. Further, it is used IO branched control loop with measurement of disturbance variable for ensuring approximate invariance of control loop. The approach used for ensuring invariance of control loop is also used to ensure decoupling control loop. imulation verification of the control method of control was presented on the two-variable control loop. The proposed approach to control is valid under the following condition, i.e. the proposed control method is considered for multi-variable controlled plant with same number input and output variables. The future wor will be focused on simulation verification of the control method for simulation model of the concrete multi-variable controlled plant. Acnowledgement This wor was supported by the European egional Development Fund under the proect CEBIA-Tech No. CZ..5/../3.89. eferences: [] J. Balate, Automatic Control, nd edition, BEN, Praha, 4, 664 pp. (in Czech) [] K. H. Johansson, The Quadruple-Tan Process: A Multivariable Laboratory Process with an Adustable Zero, IEEE Transactions on Control ystems technology, Vol. 8, No. 3,, pp [3] G. C. Goodwin,. F. Graebe and M. E. algado, Control ystem Design. Prentice Hall, Upper addle ive, New Jersey,, 98 pp. [4] F. G. hinsey, Controlling Multivariable Processes, Instrument ociety of America, esearch Triangle Par, NC, 98, 4 pp. [5] M. Åesson, E. Gustafson and K. H. Johansson, Control Design for a Helicopter Lab Process, In: Preprints 3th World Congress of IFAC, an Francisco, 996. [6] M. Mansour and W. chaufelberger, oftware and Laboratory Experiment Using Computers in Control Education, IEEE Control ystem Magazine, Vol. 9, No. 3, 989, pp [7] M. Kubalci, V. Bobal, Adaptive Control of Three-Tan-ystem Using Polynomial Methods, In: International Federation of Automatic Control, Proceedings of the 7th IFAC World Congress, oul, 8, pp [8] O. Beucher and M. Wees, Introduction to MATLAB and IMULINK, 3rd edition, Infinity cience Press LLC, Hingham, MA, 8. [9] V. Bobal, P. Chalupa, M. Kubalci and P. Dostal, elf-tuning Predictive Control of Nonlinear ervo-motor, Journal of Electrical Engineering, Vol. 6, No. 6,, pp [] A. Dastfan, Implementation and Assessment of Interactive Power Electronics Course, WEA Transactions on Advances in Engineering Education, Vol.4, No. 8, 7, pp [] E. Bristol, On a new measure of interaction for multivariable process control, IEEE Transactions on Automatic Control, Vol., No., 966, pp [] T. Glad, L. Lung, Control Theory: Multivariable and Nonlinear Methods. Taylor and Francis, NY,, 467 pp. [3] K. Warwic and D. ees, Industrial Digital Control ystems, nd edition. Institution Of Engineering And Technology, 988. [4] M. T. Tham, An Introduction to Decoupling Control, University Newcastle upon Tyne, 999. [5] J. Zaucia, Control of the Quadruple-Tan Process, CTU in Prague, diploma wor (in Czech). [6] P. Klan and. Gorez, Balanced Tuning of PI Controllers. European Journal of Control, Vol. 6, No. 6,, pp [7] M. Vitecova, A. Vitece, Bases of automatic control, nd edition, VŠB-Technical university Ostrava, Ostrava, 8. (in Czech) IBN: