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Analysis of Control System Models with Conventional LQR and Fuzzy LQR Controller
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Procedia online Computer 00 (2019) 000–000 Available at Science www.sciencedirect.com Procedia Computer Science 00 (2019) 000–000
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Procedia Computer Science 150 (2019) 737–742
13th International Symposium “Intelligent Systems” (INTELS’18) 13th International Symposium “Intelligent Systems” (INTELS’18)
Analysis of Control System Models with Conventional LQR and Analysis of Control System Models with Conventional LQR and Fuzzy LQR Controller Fuzzy LQR Controller Y.I. Kudinova,a,*, F.F. Pashchenkobb, A.Y. Kelinaaa, a Y.I. F.F. Pashchenko , A.Y. Kelinab, D.I. Kudinov Vasutinaa, *, E.S. Duvanov a, A.F. Pashchenkob D.I. Vasutin , E.S. Duvanov , A.F. Pashchenko
a Lipetsk State Technical University, Moskovskaya str., 30, Lipetsk 398600, Russia a Lipetsk State Technical University, str., 30, Lipetsk Russia Institute of Control Sciences of Russian Academy Moskovskaya of Sciences, Profsouznaya str.,398600, 65, Moscow 117997, Russia b Institute of Control Sciences of Russian Academy of Sciences, Profsouznaya str., 65, Moscow 117997, Russia b
Abstract Abstract Linear quadratic LQR and fuzzy FLC+LQR controllers are analyzed. It is shown that under the same conditions, the fuzzy Linear quadratic LQRhas and fuzzyhigher FLC+LQR controllers arecontroller. analyzed. It is shown that under the same conditions, the fuzzy FLC+LQR controller a much speed than the LQR FLC+LQR controller has a much higher speed than the LQR controller. © 2019 The Author(s). Published by Elsevier B.V. © 2019 2019 The The Author(s). Authors. Published by Elsevier B.V. © Published Elsevier B.V. This is an open access article underbythe CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee Peer-review under responsibility of the scientific committeeof ofthe the 13th 13th International InternationalSymposium Symposium“Intelligent “IntelligentSystems” Systems”(INTELS’18) (INTELS’18). Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18) Keywords: linear quadratic LQR; fuzzy FLC+LQR; controller; analysis. Keywords: linear quadratic LQR; fuzzy FLC+LQR; controller; analysis.
1. Introduction 1. Introduction In the last decade, in various industries and especially in the heat power industry, such modern control methods In the last decade,Regulation in various (LQR) industries andLQR, especially in the power industry, –such modern control methods as Linear Quadratic [1,2], together withheat linear proportional integral – differential (PID) as Linear Quadratic Regulation [1,2], LQR, together with linear proportional integral differential controllers (PID + LQR), as well(LQR) as fuzzy (FLC, Fuzzy Logic Controller) controllers–have been–widely used (PID) [3,4]. controllers (PID operating + LQR), as well as fuzzy Fuzzy Controller) have been widely used [3,4]. Under the same conditions, these(FLC, methods can Logic be arranged in thecontrollers following order: linear PID, LQR, FLC Under the+same these of methods can be arranged inInthe following order: lineartoPID, LQR, FLC and PID LQRoperating regulationconditions, as the quality regulation is improved. this paper we propose investigate and and PID the + LQR regulation as the quality ofFLC regulation improved. In this paper we propose to investigate and compare quality of transients in LQR and + LQRisregulation. compare the quality of transients in LQR and FLC + LQR regulation.
* Corresponding author. Tel.: +7-900-592-0559 * E-mail Corresponding Tel.: +7-900-592-0559 address:author. [email protected] E-mail address: [email protected] 1877-0509 © 2019 The Author(s). Published by Elsevier B.V. 1877-0509 © 2019 The article Author(s). by Elsevier license B.V. (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access underPublished the CC BY-NC-ND This is an open access article under CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18) Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18)
1877-0509 © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18). 10.1016/j.procs.2019.02.007
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Y.I. Kudinov et al. / Procedia Computer Science 150 (2019) 737–742 Y.I. Kudinov et al. / Procedia Computer Science 00 (2019) 000–000
2. Formulation of the LQR optimization problem and its classical solution The basis of the synthesis of LQR controller, which is the optimal regulator, is the method of solving the matrix Riccati equation. Suppose there is a linear continuous object, which can be described by a vector-matrix equation of the following form [5]: x (t ) Ax(t ) Bu (t ), x(t0 ) x0 ,
(1)
where x R n is the object coordinate vector, A R nn is the object parameters matrix, B R nm is the control matrix, u R m is the control vector. If a continuous object is represented by a system of matrix equations
x (t ) Ax(t ) Bu (t ), y (t ) Cx(t ) Du (t ),
(2)
then in the equation y(t) matrix A is the system matrix, B is the input matrix, C is the control matrix and D is the end – to – end matrix. An example is shown in Fig. 1, in which the control object circuit is represented by an oscillating link, and the state-Space block in the scheme is a control object described by equations (1) or (2) in the tate space and contains matrices A, B, C, D.
Fig. 1. Control object in Simulink.
The criterion of optimality of LQR control is the expression
J (u (t )) min [ xT Qx u T Ru ]dt ,
(3)
0
where Q and R are symmetric positive semidefinite matrices using the state vector x and the input vector u, respectively. In such a case, LQR is an algorithm that minimizes the optimality criterion (3) in the presence of an equality type constraint – the object model (2). To obtain the matrix of optimal feedback coefficients it is necessary to solve the matrix Riccati equation AT S SA SBR 1 B T S Q 0,
i. e. determine the optimal value of S * square positive definite symmetric matrix S . Then the optimal control can be set in the following form: u * (t ) K * x (t ) ,
where K * R1 BT S * .
Y.I. Kudinov et al. / Procedia Computer Science 150 (2019) 737–742 Y.I. Kudinov et al. / Procedia Computer Science 00 (2019) 000–000
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3. Description of model automatic control systems with traditional LQR controller Build in Matlab circuit automatic control system (ACS) (Fig. 2) containing an object described by a system of equations (2) in the state space and a traditional LQ controller.
Fig. 2. Scheme of the ACS with a traditional LQR controller.
Control object having a transfer function Wo ( s )
1 2
s 5s 3
,
(4)
it is presented in the state space by the model (2) and the state-space block, in which the matrices A, B, C, D are defined by the tf2ss function in the program1 written in the MATLAB command line. In addition, in the program in Fig. 3, it is provided to check the controllability and observability of ctrb and obsv functions and to determine in LQR the feedback coefficients K by the LQR function.
Fig. 3. Program for calculating of K.
In the scheme in Fig. 2 there is a so-called "white noise" produced by the Band-Limited White Noise unit and passing through the Fcn Transfer unit with a transfer function
Wv ( s )
10 . s 10
740 4
Y.I. Kudinov et al. / Procedia Computer Science 150 (2019) 737–742 Y.I. Kudinov et al. / Procedia Computer Science 00 (2019) 000–000
The traditional LQR controller is represented by an LQR unit. The Check Step Response Characteristics optimizer – block determines the value of the Ku1 coefficient, at which the optimal transient process will be obtained in the system, satisfying the following restrictions: the absence of overshoot and static error. When applying a stepwise perturbation and start the optimizer in the window of the Seor, we obtain Fig. 4 in the form of a continuous line diagram of the transition process in the ACS with a traditional LQR controller. The result of the optimization will receive a gain of Ku1 = 3.8.
Fig. 4. Graphs of transient processes.
4. Description of the model systems with fuzzy FLC +LQR controller We proceed to the construction of the ACS object, described by the system of equations (4) in the state space, with fuzzy FLC and linear – quadratic LQR regulators, forming the so-called FLC + LQR fuzzy controller. Let's start with a brief description of the used fuzzy controller FLC, shown in Fig. 5. The fuzzy discrete controller has two inputs: control error e(k ) and its change de(k ) e(k ) e(k 1) and one output – control u (k ) at time moments kdt , k 1, 2, ..., N , where dt is the sampling step [6].
Fig. 5. Scheme of fuzzy logic controller FLC.
Valid values of inputs e, de using normalizing coefficients K e , K de to normalized eˆ, deˆ [ 1,1], then, the operations of Fuzzyfication (Fuz), Fuzzy Inference (FI) and Defuzzyfication (Def). Operation of Fuz converts the normalized inputs eˆ, deˆ in the fuzzy E, dE. FI, for example the Mamdani method, finds fuzzy the output of dU on the basis of the fuzzy inputs E, dE and rule base RB of the form
Rc : if eˆ is E , deˆ is dE , then uˆ is dU , 1, 2, ..., q,
(5)
Y.I. Kudinov al. / Procedia Computer Science 150 (2019) 737–742 Y.I. Kudinov et al. et / Procedia Computer Science 00 (2019) 000–000
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where E , dE , dU are fuzzy sets having term – sets Te , Tde , Tdu with elements characterizing the values of the corresponding variables eˆ, deˆ (N is Negative, Z is Zero, P is Positive) and duˆ (NB is Negative Big, NM is Negative Medium, ZE is close to ZEro, PM is Positive Medium, PB is Positive Big). Operation of Def output converts the fuzzy output U, for example, the method of the median (Bisector) or center 1, 2, which is multiplied by the coefficient Kdu becomes of gravity (Centroid), normalized value uˆ [ L; L], L valid u. Below at Fig. 6 the triangular membership functions (MF) of the input fuzzy sets N, Z, P characterizing the normalized error values eˆ and its changes deˆ, marked in Matlab as e1 and de1 are given.
Fig. 6. Input MF for e1 and de1.
Membership functions of the output fuzzy sets (Fig. 7) NB, NM, ZE, PM, PB, characterize the normalized control value dû denoted in Matlab as u1.
Fig. 7. Output MF for u1.
Below at Fig. 8 fuzzy rules implemented in Matlab are given.
Fig. 8. Fuzzy rules.
The model systems with fuzzy FLC + LQR controller is shown in Fig. 9. It is obtained by adding to the diagram in Fig. 2 blocks of coefficients Ke, Kde, Kdu-scaling signals the error e, error change de and control du, blocks,
Y.I. Kudinov et al. / Procedia Computer Science 150 (2019) 737–742 Y.I. Kudinov et al. / Procedia Computer Science 00 (2019) 000–000
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delays signals in one time interval of Unit Delay, analog-to-digital Zero – Order Hold and digital-to-analog Zero – Order Hold1 converters and fuzzy controller Fuzzy Logic Controller with Ruleviewer.
Fig. 9. Scheme of the ACS with fuzzy FLC + LQR controller.
The unit Delay block and the adder calculate the change in error de(k ) e(k ) e(k 1), and the Unit Delay block 1 and the adder control uˆ (k ) duˆ (k ) uˆ (k 1) on the k -th period. When applying a step perturbation in the scheme in Fig. 9 the transition graph shown by the dashed line in Fig. 4. The analysis of transient processes obtained in control systems with LQR and FLC + LQR regulators is given in the following Table 1. Table 1. The transient analysis of control systems. Type of regulator
Coefficients gains
Process characteristic
Transition time, (с)
Rise time, (с)
–
Stable
5.5
4
7.4
Stable
5.2
1
Ku1
Ku2
LQR
3.8
FLC + LQR
6.8
As seen in Fig. 4 and Table 1, both regulators have no overshoot and static error. Both controllers provide a short transition and rise time, therefore, we can talk about the high quality of control in systems with these controllers. However, the fuzzy FLC + LQR controller has a much higher performance. Acknowledgements
Research supported by Russian Science Foundation, project # 14-19-01772. References [1] Narayana VG, Ramesh P. Modelling and Control of Single Link Manipulators for Flexible Operation by using Linearization Techniques. Int. J. of Currt Eng and Techn 2013;3:2:611-616. [2] Ganesh V, Vasu K, Bhavana P. LQR Based Load Frequency Controller for Two Area Power System. Int J. of Adv Res in Elect, Elect Instr Eng 2012;1:262-269. [3] Kumari N, Jha AN. Automatic Generation Control Using LQR based PI Controller for Multi Area Interconnected Power System. Adv in Elect and Electric Engineering 2014;4:2:149-154. [4] Thermal Power Plant Simulation and Control. Edited by D. Flynn. The Inst of Elect Eng, London; 2003. [5] Vassilyev SN, Kudinov IY, Kudinov YI, Pashchenko FF. and etc. Logical linguistic controllers, Proc Comp Sci 2017;103:629 – 636 [6] Kudinov YI, Pashchenko AF, Pashchenko FF, Arakelyan EK. Optimization of Fuzzy Parallel Settings of PID Mamdani Controller, 11th International Conference on APPL INFORM COMM TECHN. 2017; Moscow, Russia.
ScienceDirect ScienceDirect
Procedia online Computer 00 (2019) 000–000 Available at Science www.sciencedirect.com Procedia Computer Science 00 (2019) 000–000
ScienceDirect
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Procedia Computer Science 150 (2019) 737–742
13th International Symposium “Intelligent Systems” (INTELS’18) 13th International Symposium “Intelligent Systems” (INTELS’18)
Analysis of Control System Models with Conventional LQR and Analysis of Control System Models with Conventional LQR and Fuzzy LQR Controller Fuzzy LQR Controller Y.I. Kudinova,a,*, F.F. Pashchenkobb, A.Y. Kelinaaa, a Y.I. F.F. Pashchenko , A.Y. Kelinab, D.I. Kudinov Vasutinaa, *, E.S. Duvanov a, A.F. Pashchenkob D.I. Vasutin , E.S. Duvanov , A.F. Pashchenko
a Lipetsk State Technical University, Moskovskaya str., 30, Lipetsk 398600, Russia a Lipetsk State Technical University, str., 30, Lipetsk Russia Institute of Control Sciences of Russian Academy Moskovskaya of Sciences, Profsouznaya str.,398600, 65, Moscow 117997, Russia b Institute of Control Sciences of Russian Academy of Sciences, Profsouznaya str., 65, Moscow 117997, Russia b
Abstract Abstract Linear quadratic LQR and fuzzy FLC+LQR controllers are analyzed. It is shown that under the same conditions, the fuzzy Linear quadratic LQRhas and fuzzyhigher FLC+LQR controllers arecontroller. analyzed. It is shown that under the same conditions, the fuzzy FLC+LQR controller a much speed than the LQR FLC+LQR controller has a much higher speed than the LQR controller. © 2019 The Author(s). Published by Elsevier B.V. © 2019 2019 The The Author(s). Authors. Published by Elsevier B.V. © Published Elsevier B.V. This is an open access article underbythe CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee Peer-review under responsibility of the scientific committeeof ofthe the 13th 13th International InternationalSymposium Symposium“Intelligent “IntelligentSystems” Systems”(INTELS’18) (INTELS’18). Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18) Keywords: linear quadratic LQR; fuzzy FLC+LQR; controller; analysis. Keywords: linear quadratic LQR; fuzzy FLC+LQR; controller; analysis.
1. Introduction 1. Introduction In the last decade, in various industries and especially in the heat power industry, such modern control methods In the last decade,Regulation in various (LQR) industries andLQR, especially in the power industry, –such modern control methods as Linear Quadratic [1,2], together withheat linear proportional integral – differential (PID) as Linear Quadratic Regulation [1,2], LQR, together with linear proportional integral differential controllers (PID + LQR), as well(LQR) as fuzzy (FLC, Fuzzy Logic Controller) controllers–have been–widely used (PID) [3,4]. controllers (PID operating + LQR), as well as fuzzy Fuzzy Controller) have been widely used [3,4]. Under the same conditions, these(FLC, methods can Logic be arranged in thecontrollers following order: linear PID, LQR, FLC Under the+same these of methods can be arranged inInthe following order: lineartoPID, LQR, FLC and PID LQRoperating regulationconditions, as the quality regulation is improved. this paper we propose investigate and and PID the + LQR regulation as the quality ofFLC regulation improved. In this paper we propose to investigate and compare quality of transients in LQR and + LQRisregulation. compare the quality of transients in LQR and FLC + LQR regulation.
* Corresponding author. Tel.: +7-900-592-0559 * E-mail Corresponding Tel.: +7-900-592-0559 address:author. [email protected] E-mail address: [email protected] 1877-0509 © 2019 The Author(s). Published by Elsevier B.V. 1877-0509 © 2019 The article Author(s). by Elsevier license B.V. (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access underPublished the CC BY-NC-ND This is an open access article under CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18) Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18)
1877-0509 © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18). 10.1016/j.procs.2019.02.007
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Y.I. Kudinov et al. / Procedia Computer Science 150 (2019) 737–742 Y.I. Kudinov et al. / Procedia Computer Science 00 (2019) 000–000
2. Formulation of the LQR optimization problem and its classical solution The basis of the synthesis of LQR controller, which is the optimal regulator, is the method of solving the matrix Riccati equation. Suppose there is a linear continuous object, which can be described by a vector-matrix equation of the following form [5]: x (t ) Ax(t ) Bu (t ), x(t0 ) x0 ,
(1)
where x R n is the object coordinate vector, A R nn is the object parameters matrix, B R nm is the control matrix, u R m is the control vector. If a continuous object is represented by a system of matrix equations
x (t ) Ax(t ) Bu (t ), y (t ) Cx(t ) Du (t ),
(2)
then in the equation y(t) matrix A is the system matrix, B is the input matrix, C is the control matrix and D is the end – to – end matrix. An example is shown in Fig. 1, in which the control object circuit is represented by an oscillating link, and the state-Space block in the scheme is a control object described by equations (1) or (2) in the tate space and contains matrices A, B, C, D.
Fig. 1. Control object in Simulink.
The criterion of optimality of LQR control is the expression
J (u (t )) min [ xT Qx u T Ru ]dt ,
(3)
0
where Q and R are symmetric positive semidefinite matrices using the state vector x and the input vector u, respectively. In such a case, LQR is an algorithm that minimizes the optimality criterion (3) in the presence of an equality type constraint – the object model (2). To obtain the matrix of optimal feedback coefficients it is necessary to solve the matrix Riccati equation AT S SA SBR 1 B T S Q 0,
i. e. determine the optimal value of S * square positive definite symmetric matrix S . Then the optimal control can be set in the following form: u * (t ) K * x (t ) ,
where K * R1 BT S * .
Y.I. Kudinov et al. / Procedia Computer Science 150 (2019) 737–742 Y.I. Kudinov et al. / Procedia Computer Science 00 (2019) 000–000
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3. Description of model automatic control systems with traditional LQR controller Build in Matlab circuit automatic control system (ACS) (Fig. 2) containing an object described by a system of equations (2) in the state space and a traditional LQ controller.
Fig. 2. Scheme of the ACS with a traditional LQR controller.
Control object having a transfer function Wo ( s )
1 2
s 5s 3
,
(4)
it is presented in the state space by the model (2) and the state-space block, in which the matrices A, B, C, D are defined by the tf2ss function in the program1 written in the MATLAB command line. In addition, in the program in Fig. 3, it is provided to check the controllability and observability of ctrb and obsv functions and to determine in LQR the feedback coefficients K by the LQR function.
Fig. 3. Program for calculating of K.
In the scheme in Fig. 2 there is a so-called "white noise" produced by the Band-Limited White Noise unit and passing through the Fcn Transfer unit with a transfer function
Wv ( s )
10 . s 10
740 4
Y.I. Kudinov et al. / Procedia Computer Science 150 (2019) 737–742 Y.I. Kudinov et al. / Procedia Computer Science 00 (2019) 000–000
The traditional LQR controller is represented by an LQR unit. The Check Step Response Characteristics optimizer – block determines the value of the Ku1 coefficient, at which the optimal transient process will be obtained in the system, satisfying the following restrictions: the absence of overshoot and static error. When applying a stepwise perturbation and start the optimizer in the window of the Seor, we obtain Fig. 4 in the form of a continuous line diagram of the transition process in the ACS with a traditional LQR controller. The result of the optimization will receive a gain of Ku1 = 3.8.
Fig. 4. Graphs of transient processes.
4. Description of the model systems with fuzzy FLC +LQR controller We proceed to the construction of the ACS object, described by the system of equations (4) in the state space, with fuzzy FLC and linear – quadratic LQR regulators, forming the so-called FLC + LQR fuzzy controller. Let's start with a brief description of the used fuzzy controller FLC, shown in Fig. 5. The fuzzy discrete controller has two inputs: control error e(k ) and its change de(k ) e(k ) e(k 1) and one output – control u (k ) at time moments kdt , k 1, 2, ..., N , where dt is the sampling step [6].
Fig. 5. Scheme of fuzzy logic controller FLC.
Valid values of inputs e, de using normalizing coefficients K e , K de to normalized eˆ, deˆ [ 1,1], then, the operations of Fuzzyfication (Fuz), Fuzzy Inference (FI) and Defuzzyfication (Def). Operation of Fuz converts the normalized inputs eˆ, deˆ in the fuzzy E, dE. FI, for example the Mamdani method, finds fuzzy the output of dU on the basis of the fuzzy inputs E, dE and rule base RB of the form
Rc : if eˆ is E , deˆ is dE , then uˆ is dU , 1, 2, ..., q,
(5)
Y.I. Kudinov al. / Procedia Computer Science 150 (2019) 737–742 Y.I. Kudinov et al. et / Procedia Computer Science 00 (2019) 000–000
7415
where E , dE , dU are fuzzy sets having term – sets Te , Tde , Tdu with elements characterizing the values of the corresponding variables eˆ, deˆ (N is Negative, Z is Zero, P is Positive) and duˆ (NB is Negative Big, NM is Negative Medium, ZE is close to ZEro, PM is Positive Medium, PB is Positive Big). Operation of Def output converts the fuzzy output U, for example, the method of the median (Bisector) or center 1, 2, which is multiplied by the coefficient Kdu becomes of gravity (Centroid), normalized value uˆ [ L; L], L valid u. Below at Fig. 6 the triangular membership functions (MF) of the input fuzzy sets N, Z, P characterizing the normalized error values eˆ and its changes deˆ, marked in Matlab as e1 and de1 are given.
Fig. 6. Input MF for e1 and de1.
Membership functions of the output fuzzy sets (Fig. 7) NB, NM, ZE, PM, PB, characterize the normalized control value dû denoted in Matlab as u1.
Fig. 7. Output MF for u1.
Below at Fig. 8 fuzzy rules implemented in Matlab are given.
Fig. 8. Fuzzy rules.
The model systems with fuzzy FLC + LQR controller is shown in Fig. 9. It is obtained by adding to the diagram in Fig. 2 blocks of coefficients Ke, Kde, Kdu-scaling signals the error e, error change de and control du, blocks,
Y.I. Kudinov et al. / Procedia Computer Science 150 (2019) 737–742 Y.I. Kudinov et al. / Procedia Computer Science 00 (2019) 000–000
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delays signals in one time interval of Unit Delay, analog-to-digital Zero – Order Hold and digital-to-analog Zero – Order Hold1 converters and fuzzy controller Fuzzy Logic Controller with Ruleviewer.
Fig. 9. Scheme of the ACS with fuzzy FLC + LQR controller.
The unit Delay block and the adder calculate the change in error de(k ) e(k ) e(k 1), and the Unit Delay block 1 and the adder control uˆ (k ) duˆ (k ) uˆ (k 1) on the k -th period. When applying a step perturbation in the scheme in Fig. 9 the transition graph shown by the dashed line in Fig. 4. The analysis of transient processes obtained in control systems with LQR and FLC + LQR regulators is given in the following Table 1. Table 1. The transient analysis of control systems. Type of regulator
Coefficients gains
Process characteristic
Transition time, (с)
Rise time, (с)
–
Stable
5.5
4
7.4
Stable
5.2
1
Ku1
Ku2
LQR
3.8
FLC + LQR
6.8
As seen in Fig. 4 and Table 1, both regulators have no overshoot and static error. Both controllers provide a short transition and rise time, therefore, we can talk about the high quality of control in systems with these controllers. However, the fuzzy FLC + LQR controller has a much higher performance. Acknowledgements
Research supported by Russian Science Foundation, project # 14-19-01772. References [1] Narayana VG, Ramesh P. Modelling and Control of Single Link Manipulators for Flexible Operation by using Linearization Techniques. Int. J. of Currt Eng and Techn 2013;3:2:611-616. [2] Ganesh V, Vasu K, Bhavana P. LQR Based Load Frequency Controller for Two Area Power System. Int J. of Adv Res in Elect, Elect Instr Eng 2012;1:262-269. [3] Kumari N, Jha AN. Automatic Generation Control Using LQR based PI Controller for Multi Area Interconnected Power System. Adv in Elect and Electric Engineering 2014;4:2:149-154. [4] Thermal Power Plant Simulation and Control. Edited by D. Flynn. The Inst of Elect Eng, London; 2003. [5] Vassilyev SN, Kudinov IY, Kudinov YI, Pashchenko FF. and etc. Logical linguistic controllers, Proc Comp Sci 2017;103:629 – 636 [6] Kudinov YI, Pashchenko AF, Pashchenko FF, Arakelyan EK. Optimization of Fuzzy Parallel Settings of PID Mamdani Controller, 11th International Conference on APPL INFORM COMM TECHN. 2017; Moscow, Russia.