- Email: [email protected]
Adaptive Robust Stabilization of an Aperiodic Transient Process Control Quality in Systems with Interval Parametric Uncertainty
17th IFAC Workshop on Control Applications of Optimization 17th IFAC Workshop onOctober Control 15-19, Applications Yekaterinburg, Russia, 2018 of Optimization 17th IFAC Workshop Workshop onOctober Control 15-19, Applications ofonline Optimization Available at www.sciencedirect.com 17th IFAC on Control Applications Optimization Yekaterinburg, Russia, 2018 of 17th IFAC Workshop onOctober Control 15-19, Applications Yekaterinburg, Russia, 2018 Yekaterinburg, Russia, October 15-19, 2018 of Optimization Yekaterinburg, Russia, October 15-19, 2018
ScienceDirect
IFAC PapersOnLine 51-32 (2018) 826–831
Adaptive Robust Stabilization of an Aperiodic Transient Process Control Quality Adaptive of an Transient Process Adaptive Robust Robust Stabilization Stabilization ofInterval an Aperiodic Aperiodic Transient Process Control Control Quality Quality Systems with Parametric Uncertainty Adaptive Robustin Stabilization of an Aperiodic Transient Process Control Quality in Systems with Interval Parametric Uncertainty in Systems with Interval Parametric Uncertainty in Systems with Interval Uncertainty Ivan V. Parametric Khozhaev*
Ivan V. Khozhaev* Ivan Ivan V. V. Khozhaev* Khozhaev* Ivan V. Khozhaev* *Division for Automation and Robotics, National Research Tomsk Polytechnic University, *Division for Automation and Robotics, National Research Tomsk Polytechnic University, Tomsk, Russia, (e-mail: [email protected]) *Division for for Automation Automation and and Robotics, Robotics, National National Research Research Tomsk Polytechnic Polytechnic University, University, *Division Tomsk Tomsk, Russia, (e-mail: [email protected]) *Division for Automation and Robotics, Tomsk Polytechnic University, Tomsk, Russia,National (e-mail:Research [email protected]) Tomsk, Russia, (e-mail: [email protected]) Tomsk, Russia, (e-mail: [email protected]) Abstract: The paper is dedicated to a development of a controller synthesis method for system with Abstract: The paper isDeveloped dedicated to a development of aapproaches controller synthesis for system with uncertainty parameters. method combines two to control:method an adaptive one and an Abstract: The The paper is is dedicated dedicated to to development of aa controller controller synthesis synthesis method for system system with Abstract: paper aa development of method for with uncertainty parameters. Developed method combines two approaches to control: an adaptive one and an Abstract: The paper is dedicated to a development of a controller synthesis method for system with interval one;parameters. and allowsDeveloped providing method a desired quality two of an aperiodic totransient process despite interval uncertainty combines approaches control: an adaptive one and uncertainty parameters. Developed method combines two approaches totransient control:process an adaptive oneinterval and an an interval one; and allows providing a desired quality of of an aperiodic despite uncertainty parameters. Developed method combines two approaches to control: an adaptive one and an parametric uncertainty of a system. Application example a developed method is provided; synthesized interval one; and allows providing aa desired quality of an aperiodic transient process despite interval interval one; and allows providing desired quality of an aperiodic transient process despite interval parametric uncertainty of aproviding system. Application example of aaperiodic developed method is provided; synthesized interval one; and allows a desired quality of an transient process despite interval system was examined via simulation modeling. Synthesis results, obtained with the help of developed parametric uncertainty of aa system. example of aa developed method is provided; synthesized parametric uncertainty of system. Application Application example of results, developed method provided; system was examined via simulation modeling. Synthesis obtained withis help ofsynthesized developed parametric uncertainty system.obtained Application of results, a developed method is the provided; synthesized method, were comparedof toa results, withexample previously developed methods. system was examined via simulation modeling. Synthesis obtained with the help of system was examined via simulation modeling. Synthesis results, obtained with the help of developed developed method, were compared to results, obtained with previously developed methods. system was examined via simulation modeling. Synthesis results, obtained with the help of developed method, were compared to results, results, obtained with previously developed methods. method, were compared to obtained with previously developed methods. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: robust control, adaptive control, parametric uncertainty, aperiodic transient process, control method, were compared to results, obtained with previously developed methods. Keywords: robustsimulation control, adaptive control, parametric uncertainty, aperiodic transient process, control system synthesis, modeling. Keywords: robust control, control, Keywords: robustsimulation control, adaptive adaptive control, parametric parametric uncertainty, uncertainty, aperiodic aperiodic transient transient process, process, control control system synthesis, modeling. Keywords: robust control, adaptive control, parametric uncertainty, aperiodic transient process, control system system synthesis, synthesis, simulation simulation modeling. modeling. system synthesis, simulation modeling. required data, simple structure of a controller, capability to 1. INTRODUCTION required data, simple structurevariations of a controller, to compensate rapid parametric in widecapability ranges and 1. INTRODUCTION required simple structure of capability to required data, data, simple structurevariations of aa controller, controller, capability to compensate rapid parametric in wide ranges and 1. INTRODUCTION required data, simple structure of a controller, capability to 1. INTRODUCTION stability of control quality indices. This is the main aim of the A vast variety of control objects or systems are known to compensate rapid parametric variations in wide ranges and compensate rapid parametric variations in wide ranges and 1. INTRODUCTION stability of control quality indices. This is the main aim of the A vast variety of control objects or systems are known to compensate rapid parametric variations in wide ranges and paper: to develop a method of controller synthesis with have parametric uncertainty. Such uncertainty must be taken stability of of control control quality quality indices. indices. This This is is the the main main aim aim of of the the A vast variety or are to A vast variety of of control control objects objects or systems systems must are known known to stability paper: toof develop a method of This controller synthesis have parametric Such uncertainty be taken control quality indices. is thethe main ofwith the aforementioned features. In order to reach aim,aim a set of A vast variety ofuncertainty. control or of systems aresystems known to stability into consideration during objects synthesis controlmust for paper: to develop a method of controller synthesis with paper: to develop a method of controller synthesis with have parametric uncertainty. Such uncertainty be taken have consideration parametric uncertainty. Such uncertainty must be taken features. In order to reach the aim, a set of into during synthesis of control systems for aforementioned paper: to develop a method of controller synthesis with problems must be solved: a problem of the paper must have parametric uncertainty. Such uncertainty must be taken sophisticated robotics, such as industrial robots, unmanned aforementioned features. features. In In order order to to reach reach the the aim, aim, aa set set of of into consideration during synthesis of systems for into consideration during synthesis of control control systems for aforementioned problems must be solved: a problem of the paper must sophisticated robotics, such as industrial robots, unmanned aforementioned Inoforder to reach aim, a set of formulated,must an features. algorithm designing anthe adaptive-robust into consideration during synthesis of control systems for problems aerial or underwater vehicles, etc. There arerobots, several common be solved: a problem of the paper must problems must be solved: a problem of the paper must sophisticated robotics, such as industrial unmanned sophisticated robotics, such asetc. industrial unmanned formulated,must an be algorithm designing an the adaptive-robust aerial or underwater vehicles, There arerobots, several common controller problems bedeveloped, solved:of a the problem of paper must algorithm must be applied sophisticated robotics, such as industrial robots, unmanned approaches to the synthesis of control systems with uncertain formulated, an algorithm of designing an adaptive-robust formulated,must an be algorithm of the designing an must adaptive-robust aerial or or underwater underwater vehicles, vehicles, etc. etc. There There are are several several common common controller aerial developed, algorithm be system, applied approaches to the synthesis of neural control systems with formulated, anofbe algorithm of athe designing anfor adaptive-robust a problem synthesizing controller some aerial or underwater vehicles, There are several common parameters: adaptive control; networks baseduncertain control; to controller must developed, algorithm must be applied controller must be developed, the algorithm must be system, applied approaches to the synthesis synthesis of etc. control systems with uncertain approaches to the of control systems with uncertain to a problem ofbesynthesizing athecontroller for some parameters: adaptive control; neural networks based control; controller must developed, algorithm must be applied synthesized system must be examined via simulation approaches to the synthesis of control systems with uncertain fuzzy logic based control and robust control, based on to a problem of synthesizing a controller for some system, to a problem system of synthesizing a controller forviasome system, parameters: adaptive adaptive control; control; neural neural networks networks based based control; control; synthesized parameters: must be examined simulation fuzzy logic based control and robust control, based on to a problem of synthesizing a controller for some system, modeling. Let us consider solving these problems further. parameters: control; and neural networks basedbased control; considering system parameters asrobust intervals. Each of these system must be via simulation synthesized mustsolving be examined examined via further. simulation fuzzy logic adaptive based control control control, on synthesized fuzzy logic based andasrobust control, on modeling. Letsystem us consider these problems considering system parameters intervals. Eachbased of these system mustsolving be examined via further. simulation fuzzy logic has based control andasrobust control, based on synthesized approaches its own advantages and disadvantages. For modeling. Let us consider these problems modeling. Let us consider solving these problems further. considering system parameters intervals. Each of these considering system parameters as intervals. Each of these approaches has its own advantages and disadvantages. For modeling. Let us consider solving these problems further. 2. FORMULATION OF A PROBLEM considering system parameters as intervals. Each of these example, adaptive control systems provide stable control approaches has advantages and disadvantages. For approaches has its its own own advantages and disadvantages. For FORMULATION OF A PROBLEM example,butadaptive control systems provide stable control approaches has its compensate own advantages disadvantages. For quality, cannot rapid and parametric variations in Developed 2. 2.method FORMULATION OF an A PROBLEM PROBLEM example, adaptive control systems provide stable control 2. FORMULATION OF A is based on interval approach to example, adaptive control systems provide stable control quality, butadaptive cannot compensate rapid parametric variations in Developed 2.method FORMULATION OF an A PROBLEM example, control provide control wide ranges (Ioannou 2010,systems Ioannou 2012, stable Feng 1999). is based on interval approachof toa quality, but cannot compensate rapid parametric variations in control. According to this approach, each parameter quality, but cannot compensate rapid parametric variations in Developed method is based on approach wide ranges (Ioannou 2010, systems Ioannou 2012, desired Feng 1999). method to is this based on an an interval interval approachof to toa quality, but cannot compensate rapid parametric variations in Developed Neural network based control provide control control. According approach, each parameter wide ranges (Ioannou 2010, Ioannou 2012, Feng 1999). Developed method is based on an interval approach toa control system may vary in some intervals. Each control wide ranges (Ioannou 2010, systems Ioannouprovide 2012, desired Feng control 1999). control. According to this approach, each parameter of Neural network based control control. According to this approach, each parameter of a wide ranges (Ioannou 2010, Ioannou 2012, Feng 1999). quality, but require a vast amount of data to train a neural control system may vary invia some intervals. Each control Neural network network based based control control systems systems provide provide desired desired control control control. According to this approach, each parameter of a Neural system can be described denominator of a transfer quality, but require a control vastAlso, amount ofprovide data to desired train a control neural control system may vary in some intervals. Each control control system may vary in some intervals. Each control Neural network based systems network (Wang 2009). both adaptive and neural system system can be may described viasome denominator of a transfer quality, but but require require aa vast vast amount amount of of data data to to train train aa neural neural control vary in intervals. Each control quality, function – characteristics polynomial. Let us designate network (Wang 2009). adaptive anda neural can be via of aa transfer quality, require a vastAlso, amount of data to train neural system be described described polynomial. via denominator denominator ofdesignate transferaa network but based controllers have aboth complex structure. Fuzzy system network (Wang 2009). Also, both adaptive and neural neural function can – characteristics Let uswith network (Wang 2009). Also, both adaptive and system can be described via denominator of a transfera characteristic polynomial of a system interval network based controllers have aboth complex structure. Fuzzy function – polynomial. Let designate network (Wang 2009). adaptive neural logic based controller has aAlso, simpler structure, but aand procedure function – characteristics characteristics polynomial. Let us uswith designate a network based controllers have aa complex structure. Fuzzy characteristic polynomial of a system interval network based controllers have complex structure. Fuzzy function – characteristics polynomial. Let us designate as interval characteristic polynomial (ICP). Wea logic based controller a have simpler structure, but depends a procedure characteristic polynomial of aa system with interval network based a complex structure. Fuzzy of designing acontrollers rule has base is not strict and on parameters characteristic polynomial of system with interval logic based controller has a simpler structure, but a procedure parameters interval characteristic polynomial Wea logic based controller a simpler structure, but depends a procedure polynomial of aas interval system with(ICP). interval will consideras of ICP parameters of designing rule has base is ofnotan strict and on characteristic parameters as interval polynomial (ICP). We logic based controller has a simpler structure, but(Lee a procedure experience anda qualification engineer 2001). ascoefficients interval characteristic characteristic polynomial (ICP). of Wea of rule and depends on will consider coefficients of ICP as interval parameters of of designing designingandaa qualification rule base base is is ofnot notanstrict strict and (Lee depends on parameters parameters as interval characteristic polynomial (ICP). Wea experience engineer 2001). system, so, general form ofofICP can be writtenparameters as follows:of of designing a rule base is not strict and depends on Procedures of interval control system synthesis are strictly will consider coefficients ICP as interval will consider coefficients of ICP as interval parameters of experience and qualification of an engineer (Lee 2001). experience of andinterval qualification of an engineer (Lee 2001). will system, so, general form ofofICP can be writtenparameters as follows:of aa n coefficients n ICP consider as interval Procedures control system synthesis are strictly experience and qualification of andata: engineer (Lee 2001). formulated, they requirecontrol minimum only minimal and system, general form can i of system, so, general of ICP ICP be written as as follows: follows: Procedures of of interval system synthesis are strictly strictly n q ( K ) form n q can D( s, K )so, ; qi be s ii ,written Procedures interval control system synthesis are i can i si so, general form of ICP be written as follows: (1) formulated, they require minimum data: onlydesired minimal and system, n n Procedures of interval control system synthesis are strictly maximal value of each parameter, and provide control n n D ( s, K ) (1) formulated, they they require require minimum minimum data: data: only only minimal minimal and and i n0 q ( K ) i s ii i n0 q i ; q i s ii , formulated, maximal value of each parameter, anddata: provide desired control D q( K ) s i 0 q (1) D(( ss,, K K )) qii ;; q qii ss i ,, (1) formulated, they require minimum only minimal and quality rapid parametric variations in wide ranges, but i 0 q ( K ) ii s i maximaldespite value of each each parameter, and provide provide desired control D ( s , K ) q ( K ) s q ; q s , (1) maximal value of parameter, and desired control i i 0 0 0 0 i i i quality despite rapid parametric variations in wide ranges, but i i D ( s , K ) where – is an ICP, – is a vector of controller maximal value of each parameter, and provide desired control K synthesized system has insufficient dynamics and value of its i 0 i 0 quality despite despite rapid rapid parametric parametric variations variations in in wide wide ranges, ranges, but but quality where D ( s, K ) – is an ICP, K – is a vector of controller synthesized system insufficient dynamics and value ofbut its quality indices despite rapidhas parametric invariations wide ranges, may vary during variations parametric (Polyak where aa vector controller synthesized system has insufficient dynamics and of D (( ss,, K K )) – where D – )is is an anisICP, ICP, – is is vector of of controller K synthesized system has insufficient dynamics and value value of its its parameters, an K ICP– coefficient function of quality indices may vary during parametric variations (Polyak where D ( s, Kqq i)(K – )is – an ICP, – is a vector of controller K synthesized system has insufficient dynamics and value of its 2002, Bhattacharyya 1995). So, parametric a problem of developing new parameters, quality indices may vary during variations (Polyak (K – is an ICP coefficient function of quality indices may vary during parametric variations (Polyak i 2002, Bhattacharyya 1995). So, parametric a for problem of developing new parameters, )) – coefficient function of quality indices may vary during variations (Polyak methods of controller synthesis systems with uncertain (K – is is qan ani , qICP ICP coefficient function of parameters, minimal and maximal parameters ofqqaii (K controller, 2002, Bhattacharyya 1995). So, aa problem of developing new i – are 2002, Bhattacharyya 1995). So, problem of developing new q (K ) – is an ICP coefficient function of parameters, methods ofiscontroller synthesis for systems with uncertain parameters of ai controller, q i , q i – are minimal and maximal 2002, Bhattacharyya 1995). So, a problem of developing new parameters highly relevant. methods of controller synthesis for systems with uncertain q , q – are minimal and maximal parameters of a controller, methods ofiscontroller synthesis for systems with uncertain values of each ICP coefficient, parameters of a q minimal and maximal parameters of a controller, i , qwhich i – areinclude parameters highly relevant. i i methods ofiscontroller synthesis for systems with uncertain values q i , qwhich minimal and maximal parameters of aICP controller, of each coefficient, parameters of a i – areinclude parameters relevant. parameters is highly highly relevant. One of ways to improve existing method is to combine it controller. values of each ICP coefficient, which include parameters of values of each ICP coefficient, which include parameters of aa parameters is highly relevant. controller. One of ways to improve existing method is to combine it values of each ICP coefficient, which include parameters of a with the other one. This paper is dedicated to a development controller. controller. One of of ways ways to to improve improve existing existing method method is is to to combine combine it it Roots One of (1) are located in some areas of a complex plane due with the other one. This paper is dedicated to a development controller. One of ways improve existing method is atodevelopment combine it Roots of (1) are located in some areas of a complex plane due of a combined method controller synthesis, based on an with the other to one. This of paper is dedicated dedicated to with the other one. This paper is to a development ICPof coefficients variations. This causes variation plane of values of a combined method of controller synthesis, based on an to Roots (1) in areas of due with the other one. This paper is dedicated to a development of (1) are are located located in some some areas of aa complex complex plane due adaptive approach and an interval approach simultaneously. to ICP coefficients variations. This causes variationdegree. of values of aa combined combined method method of of controller controller synthesis, synthesis, based based on on an an Roots Roots of (1) are located in some areas of a complex plane due of of quality indices: stability degree and oscillability In adaptive approach and an interval approach simultaneously. to ICP coefficients variations. This causes variation of values ICP coefficients variations. Thisand causes variationdegree. of values of a combined method ofcombine controller synthesis, an to Developed method willan advantages ofbased theseontwo of quality indices: stability degree oscillability In adaptive approach and interval approach simultaneously. to ICP coefficients variations. This causes variation of values adaptive approach and an interval approach simultaneously. order to reduce oscillation of control quality, a poles Developed method willan combine of these two of of quality quality indices: indices: stability stability degree degree and and oscillability oscillability degree. degree. In In adaptive approach and interval advantages approach simultaneously. approaches: sufficient dynamics of a system, minimum order to reduce oscillation of control quality, a poles Developed method will combine advantages of these two quality degree oscillability In Developed will dynamics combine advantages of these two of approaches: method sufficient of a system, minimum order to indices: reduce stability oscillation of and control quality,degree. a poles poles order to reduce oscillation of control quality, a Developed method will combine advantages of these two approaches: approaches: sufficient sufficient dynamics dynamics of of aa system, system, minimum minimum order to reduce oscillation of control quality, a poles approaches: sufficient dynamics Federation of a system, minimum 2405-8963 © 2018, IFAC (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2018 IFAC 826 Peer review©under of International Federation of Automatic Copyright 2018 responsibility IFAC 826Control. Copyright © 826 10.1016/j.ifacol.2018.11.443 Copyright © 2018 2018 IFAC IFAC 826 Copyright © 2018 IFAC 826
IFAC CAO 2018 Yekaterinburg, Russia, October 15-19, 2018Ivan V. Khozhaev / IFAC PapersOnLine 51-32 (2018) 826–831
dominance principle can be applied. According to this principle, in order to provide desired control quality, one or two poles of a system (dominant poles) must be placed on a complex plane near an imaginary axis; all other poles must be placed far enough away of dominant poles to minimize their influence on a transient process. An example of such allocation is shown in the figure 1.
827
through achieving stability of B ( s, K ) , despite its interval parametric uncertainty, and simultaneously.
R ( s, K )
equality to zero
3. DERIVATION OF GENERAL EXPRESSIONS Let us derive expressions, which allow calculating remainder and coefficients of non-dominant polynomial. As far, as we know a general form of ICP (1) and a dominant polynomial (3), we can derive an expression of B ( s, K ) coefficients and a remainder R ( s, K ) . These expressions can be written as follows:
bi q( K )i1 bi1 ; n
j n 1...0,
R( K ) q( K ) i i D( , K ),
(4) (5)
i 0
where
b i
– are interval coefficients of non-dominant
polynomial B ( s, K ) ; q( K ) i – are interval coefficients of ICP, which depend on parameters of a controller; – is a value of a dominant pole, which defines control quality of a
Fig. 1. Example of poles allocation of a system with interval parameters
system; R(K ) – is a remainder of ICP D( s, K ) division by dominant polynomial A(s ) . Let us formulate a synthesis algorithm on a base of (4) and (5).
Allocation of system poles, given in the figure 1, provides an aperiodic transient process, which duration is determined by a dominant pole, which lies in an interval 6;5 . This causes a variation of stability degree in an interval 5;6 . Such allocation can be provided with the help of several methods (Nesenchuk 2002, Nesenchuk 2008, Nesenchuk 2017, Wang 2002, Zhmud 2014, Zhmud 2017, Tagami 2003, Gayvoronskiy 2014, Gayvoronskiy 2015, Gayvoronskiy 2017). The aim of the paper is to improve these methods, by placing a dominant pole in a point of a complex plane, but not in an interval. This will improve stability of quality indices of a system.
4. DEVELOPMENT OF THE SYNTHESIS ALGORITHM In order to formulate the synthesis algorithm, we will define required data, algorithm itself and its resulting data. First of all, a desired control quality must be defined. The method is based on a pole dominance principle, so desired control quality is defined through defining location of dominant and non-dominant poles. Location of dominant pole corresponds to desire stability degree of a system. Location of non-dominant poles corresponds to desired oscillability degree and distance to dominant pole.
Developed method is based on decomposing ICP in three parts: a dominant polynomial A(s ) ; a non-dominant polynomial B ( s, K ) and a remainder R ( s, K ) : D( s, K ) A( s) B( s, K ) R( K ).
(2)
Desired control quality can be provided in three steps: defining A(s ) according to desired control quality; providing a stability of B ( s, K ) and providing equality of R ( s, K ) to zero. The paper is dedicated to a development of a method of providing an aperiodic transient process, so we should place one real dominant pole according to desired value of stability degree: A( s ) s ,
(3)
where – is a dominant pole. Now we can finally formulate the problem of the research: to develop a method of synthesizing parameters of a controller, which will provide a dominant pole allocation in a point of a complex plane
Fig. 2. Example of defining desired control quality of the system through placing its poles on a complex plane
827
IFAC CAO 2018 828 Yekaterinburg, Russia, October 15-19, 2018Ivan V. Khozhaev / IFAC PapersOnLine 51-32 (2018) 826–831
In the figure 2, stability degree of the system is determined by dominant pole ; robust oscillability degree is determined by sector angle of non-dominant poles allocation; invariance of control quality from non-dominant poles is provided by their maximal real part .
where – is a maximal real part of a non-dominant pole. In order to place non-dominant poles in a sector, a following substitution must be performed:
After the aim of synthesis is set, a controller structure must be chosen. The method requires a controller with at least two tunable parameters: one for stabilizing a dominant pole in a point of a complex plane and the other for placing nondominant poles in desired area of a complex plane. Author recommends using a PI-controller, if there are no special requirements to systems oscillability and it is enough to provide some minimal real part of non-dominant poles. In the other case, if non-dominant poles must be placed in a sector, as it is shown in the figure 2, then a PID-controller should be used. Using a PID-controller allows using two tunable parameters for non-dominant poles placing and makes this problem easier. PID-controller with a following transfer function will be used in all examples of the method application in the paper:
where – is a desired maximal oscillability degree of a system. Constant values of the rest of controller coefficients, chosen from a D-partition, will provide desired allocation of non-dominant poles.
s
(9)
Summarizing all aforementioned information, a synthesis algorithm can be formulated:
KI (6) , s where K P , K I – are proportional and integral coefficients of the PI-controller. Once a controller is chosen, a transfer function of the system with interval parameters can be derived. Then, its denominator – an ICP of a system in a form (1) – can be used
1. Choose desired control quality by defining stability degree and oscillability degree of the system; define a dominant pole and parameters of non-dominant poles allocation area according to a desired control quality. 2. Choose an appropriate controller; derive a transfer function of the system and its ICP in a form (1). 3. Find a remainder R(K ) with the help of (5); solve an equation R( K ) 0 and find a proportional coefficient function of other controller parameters (7).
in (5) to derive a remainders R(K ) function of controller parameters and interval parameters of a system. By equating
4. Substitute (7) to (1) and calculate coefficients of B ( s, K ) with the help of (4).
remainder R(K ) to zero and expressing a proportional coefficient of a controller in terms of other parameters of a system, a law of its adaptation to parametric uncertainty of a system can be derived. General form of this law can be written as follows in case of using a PI-controller (6): (7)
5. Substitute (8) or (9) to B ( s, K ) and derive equations of Dpartition curves; choose values of controller parameters from D-partition and substitute them to (7). 6. Finally, a controller is synthesized; it consists of an adaptation law for a proportional coefficient and constant values of all other parameters.
where q – is a vector of interval parameters of a system. After derivation of (7), it must be substituted to (1). Since coefficients of an ICP depend on one (in case of using a PIcontroller) or two (in case of using a PID-controller),
Let us consider an example of controller synthesis with the help of the method.
coefficients of non-dominant polynomial B ( s, K ) can be calculated via (4). In order to provide desired allocation of
5. EXAMPLE OF SYNTHESIS METHOD APPLICATION Let us assume that there is a problem of providing a desired control quality in a control system with interval parametric uncertainty. Transfer function of a control object of the system is given below:
non-dominant poles, which are roots of B ( s, K ) , a Dpartition method can be used. An equation of a D-partition curve can be derived by substituting s in B ( s, K ) with a border equation of a non-dominant poles allocation area. For example, in order to provide a minimal real part of nondominant poles, following substitution must be performed: s j , 0,
j , 0,
Final form of a controller, synthesized with the help of the method, consists in a law of adaptation of a proportional coefficient, which stabilizes a dominant pole in a specified point of a complex plane, and constant values of other parameters of a controller, which provide a desired allocation of non-dominant poles.
WPI (s, K P , K I ) K P
K P f ( K I , q),
1
W (s, p, q)
(8)
828
[q1 ] s [q0 ] , [ p2 ] s 2 [ p1 ] s [ p0 ]
[q1 ] [7;10], [q0 ] [15;20], [ p 0 ] 30;40, where [ p1 ] [12;15], [ p 2 ] [5;8] – are interval parameters of the system. According to synthesis algorithm, we will define a
IFAC CAO 2018 Yekaterinburg, Russia, October 15-19, 2018Ivan V. Khozhaev / IFAC PapersOnLine 51-32 (2018) 826–831
desired stability degree and define a dominant pole (1; j 0) . Also a maximal real part of non-dominant poles is defined as follows: 2 .
K P ( p, q) 16
([ p 0 ] K P [q 0 ] K I [q1 ]) s K I [q 0 ].
p0 p1 p2 ; q0 q1
K I 16 .
Besides a control object, the system has a PI-controller with a (6) transfer function. Serial connection of the control object and the controller is enclosed with a unity feedback. ICP of the system is given below: D( s, p, q, K P , K I ) [ p 2 ] s 3 ([ p1 ] K P [q1 ]) s 2
829
Poles allocation of a synthesized system is shown in the figure 3. Figure 3 shows, that the synthesis was successful: a dominant pole is stabilized in point (1; j 0) of a complex plane; all other poles are located on the left of a line Im( x) 2 .
(10)
After a transfer function is derived, its denominator can be used to derive a remainder function with the help of (5):
R( p, q, K P , K I ) [ p 2 ] 3 ([ p1 ] K P [q1 ]) 2 ([ p0 ] K P [q0 ] K I [q1 ]) K I [q0 ] [ p 2 ] [ p1 ] K P [q1 ] [ p0 ] K P [q0 ] K I [q1 ] K I [q0 ]. Dominant pole will be placed as defined, if remainder is equal to zero. Let us derive and solve this equation: R ( p, q, K P , K I ) 0; K P ( p, q, K I ) K I
p0 p1 p 2 . q0 q1
(11)
A law of adaptation for proportional coefficient of the PIcontroller (10), which will allow stabilizing a dominant pole in a point of a complex plane, is now obtained. According to a synthesis algorithm, (11) must be substituted to (10). Considering this, an ICP of the system can be written as follows: Fig. 3. Allocation of poles of the synthesized system on a complex plane
3
D( s, p, q, K I ) [5.0;8.0] s 7.0 K I 38 .400 ;10 .0 K I 0.385 s 2 22 .0 K I 78 .0;30 .0 K I 9.231 s [15 .0 K I ;20 .0 K I ].
6. SIMULATION MODELING OF A SYNTHESIZED SYSTEM
Coefficients of this form of ICP can be used to calculate
In order to examine effectiveness of the synthesized controller, simulation modeling should be performed. A model of the system was developed with the help of MATLAB software. The model is shown in the figure 4. The model includes several generators, which provide values of interval parameters of the system, joined in the Interval Parameters subsystem; a block of transfer function with variable parameters, which represent the Control Object of the system and an adaptive PI-controller, synthesized in a previous paragraph.
coefficients of non-dominant polynomial B ( s, K ) with the help of (4):
B( s, K I ) [b2 ] s 2 [b1 ] s [b0 ]; [b2 ] [5.0;8.0]; [b1 ] 7.0 K I 46.400;10.0 K I 5.385 ;
(12)
[b0 ] 3.0 K I 33.015;3.0 K I 46.015 . Equations of D-partition can be derived by performing a substitution (8) and expressing an integral coefficient K I of the controller. Intersection of stability domains, found from D-partition in each vertex of coefficients polytope of (12), provides us with constant value of K I , which allocates nondominant poles as necessary. Let us choose K I 16 and substitute it to (11). Finally, a PI-controller, which provides desired poles allocation despite parametric uncertainty of the system, can be described as follows:
Fig. 4. Matlab model of the synthesized system
829
IFAC CAO 2018 830 Yekaterinburg, Russia, October 15-19, 2018Ivan V. Khozhaev / IFAC PapersOnLine 51-32 (2018) 826–831
Values of interval parameters vary in previously defined intervals with various frequencies as shown in the figure 5. Frequencies are chosen to provide all combinations of values during a period of q1 oscillation.
in following parameters: proportional coefficient K P 2 , integral coefficient K I 2.558 and differential coefficient K D 0.123 . Step responses of two synthesized systems are shown in the figure 7.
Fig. 5. Values of interval parameters of the system
Fig. 7. Comparison of step responses of systems with an adaptive robust controller and robust controller with constant parameters
In order to perform simulation modeling of the system, a form of input signal must be chosen. Most of control systems with uncertain parameters are synthesized with an assumption that parameters of a system vary much slower, than input signal of the system. In order to examine synthesized system in more rigid conditions, we will use a sine function, which varies slower, than parameters of the system, as an input signal. Reaction of the system on such input signal is shown in the figure 6.
Figure 7 shows, that adaptive robust PI-controller is far more effective, than a robust PID-controller with constant parameters. Mean deviation between input and output signals of the system is equal to 0.967% for adaptive robust PIcontroller, synthesized with the help of proposed method, and 3.618% for robust PID-controller with constant parameters. 7. CONCLUSIONS The main aim of the research was reached: a new method of synthesis, which allows designing adaptive robust controllers, was developed. Controllers, synthesized with the help of the method, have advantages of interval and adaptive approaches to manipulating systems with uncertain parameters: a procedure of synthesis is simple; a controller structure is simple and strictly defined; a controller is able to damp rapid significant parametric disturbances and provides stable control quality. All these features were proved by simulation modelling and comparison with a controller, synthesized with the help of another method (Gayvoronskiy 2017). ACKNOWLEDGMENT The reported study is supported by the Ministry of Education and Science of Russian Federation (project #2.3649.2017/PCh).
Fig. 6. Sine input and output of the synthesized system
REFERENCES
Figure 6 shows, that the controller, synthesized with the help of the developed method, is effective in conditions of interval parametric uncertainty.
Ioannou, P., Baldi, S. Robust Adaptive Control. (2010). Levine S. W. (ed.), The Control Systems Handbook, Second Edition: Control System Advanced Methods, 35. CRC Press, Boca Raton, Florida. Ioannou, P.A., Sun, J. (2012). Robust Adaptive Control, p. 821. Courier Corporation, United States.
Let us synthesize a robust PID-controller with constant parameters for the same system with the help of method, described in (Gayvoronskiy 2017), and compare results of simulation modeling. Synthesis of the PID-controller resulted 830
IFAC CAO 2018 Yekaterinburg, Russia, October 15-19, 2018Ivan V. Khozhaev / IFAC PapersOnLine 51-32 (2018) 826–831
Feng, G., Lozano, R. (1999). Adaptive Control Systems, p. 352. Newnes, Oxford, United Kingdom. Wang, D., Huang, J., Lan, W., Li, X. (2009). Neural networkbased robust adaptive control of nonlinear systems with unmodeled dynamics. Mathematics and Computers in Simulation 79(5), pp. 1745-1753. Lee, H.J., Park, J.B., Chen, G. (2001). Robust fuzzy control of nonlinear systems with parametric uncertainties. IEEE Transactions on Fuzzy Systems, 9, № 2, pp. 369-379. Polyak, B.T, Scherbakov, P.S. (2002). Robust stability and control, p.303. Science, Moscow. Bhattacharyya, S.P, Chapellat, H., Keel L.H. (1995). Robust control: The parametric approach, p. 672. Prentice-Hall, United States Nesenchuk, A.A. (2002). Parametric synthesis of qualitative robust control systems using root locus fields. Proceedings of the 15th Triennial World Congress of The International Federation of Automatic Control (IFAC) Barcelona, Spain 21–26 July 2002. pp. 387-387. Nesenchuk, A.A., Fedorovich, S.M. (2008). Parametric synthesis method of integral systems on the basis of root locus curves of Kharitonov polynomials. Automation and Remote Control, 69 (7), pp. 1133-1141. Nesenchuk A.A. (2017). A method for synthesis of robust interval polynomials using the extended root locus. Proceedings of the American Control Conference 24 -26 May 2017, Sheraton Seattle; United States, pp. 1715172. Wang, Y., Schinkel, M., K.J. Hunt. (2002). PID and PID-like controller design by pole assignment within D-stable regions. Asian Journal of Control, 4, № 4, pp. 423-432. Zhmud, V., Dimitrov, L., Yadrishnikov, O. (2014). Calculation of regulators for the problems of mechatronics by means of the numerical optimization method. Proceedings of 12th International Conference on Actual Problems of Electronic Instrument Engineering, APEIE 2014; Novosibirsk; Russian Federation; 2 - 4 October 2014, pp. 739-744. Zhmud, V.A., Reva, I.L., Dimitrov, L.V. (2017). Design of robust systems by means of the numerical optimization with harmonic changing of the model parameters. Journal of Physics: Conference Series, 803 (1), article number 012185. Tagami, T. Ikeda, K. (2003). Design of robust pole assignment based on Pareto-optimal solutions. Asian Journal of Control, 5(2), pp. 195-205. Gayvoronskiy S.A., Ezangina T. (2014). The synthesis of the robust stabilization system of cable tension for the test bench of weightlessness simulation. Advanced Materials Research, 1016, pp. 394-399. Gayvoronskiy S.A., Ezangina T., Khozhaev I. (2015). The interval-parametric synthesis of a linear controller of speed control system of a descent submersible vehicle. IOP Conference Series: Materials Science and Engineering, 93, article number 012055. Gayvoronskiy S.A., Ezangina T., Khozhaev I. (2017). Method of interval system poles allocation based on a domination principle. International Conference on Mechanical, System and Control Engineering, ICMSC
831
2017; St. Petersburg; Russian Federation; 19 May 2017 21 May 2017, pp. 245-249.
831
ScienceDirect
IFAC PapersOnLine 51-32 (2018) 826–831
Adaptive Robust Stabilization of an Aperiodic Transient Process Control Quality Adaptive of an Transient Process Adaptive Robust Robust Stabilization Stabilization ofInterval an Aperiodic Aperiodic Transient Process Control Control Quality Quality Systems with Parametric Uncertainty Adaptive Robustin Stabilization of an Aperiodic Transient Process Control Quality in Systems with Interval Parametric Uncertainty in Systems with Interval Parametric Uncertainty in Systems with Interval Uncertainty Ivan V. Parametric Khozhaev*
Ivan V. Khozhaev* Ivan Ivan V. V. Khozhaev* Khozhaev* Ivan V. Khozhaev* *Division for Automation and Robotics, National Research Tomsk Polytechnic University, *Division for Automation and Robotics, National Research Tomsk Polytechnic University, Tomsk, Russia, (e-mail: [email protected]) *Division for for Automation Automation and and Robotics, Robotics, National National Research Research Tomsk Polytechnic Polytechnic University, University, *Division Tomsk Tomsk, Russia, (e-mail: [email protected]) *Division for Automation and Robotics, Tomsk Polytechnic University, Tomsk, Russia,National (e-mail:Research [email protected]) Tomsk, Russia, (e-mail: [email protected]) Tomsk, Russia, (e-mail: [email protected]) Abstract: The paper is dedicated to a development of a controller synthesis method for system with Abstract: The paper isDeveloped dedicated to a development of aapproaches controller synthesis for system with uncertainty parameters. method combines two to control:method an adaptive one and an Abstract: The The paper is is dedicated dedicated to to development of aa controller controller synthesis synthesis method for system system with Abstract: paper aa development of method for with uncertainty parameters. Developed method combines two approaches to control: an adaptive one and an Abstract: The paper is dedicated to a development of a controller synthesis method for system with interval one;parameters. and allowsDeveloped providing method a desired quality two of an aperiodic totransient process despite interval uncertainty combines approaches control: an adaptive one and uncertainty parameters. Developed method combines two approaches totransient control:process an adaptive oneinterval and an an interval one; and allows providing a desired quality of of an aperiodic despite uncertainty parameters. Developed method combines two approaches to control: an adaptive one and an parametric uncertainty of a system. Application example a developed method is provided; synthesized interval one; and allows providing aa desired quality of an aperiodic transient process despite interval interval one; and allows providing desired quality of an aperiodic transient process despite interval parametric uncertainty of aproviding system. Application example of aaperiodic developed method is provided; synthesized interval one; and allows a desired quality of an transient process despite interval system was examined via simulation modeling. Synthesis results, obtained with the help of developed parametric uncertainty of aa system. example of aa developed method is provided; synthesized parametric uncertainty of system. Application Application example of results, developed method provided; system was examined via simulation modeling. Synthesis obtained withis help ofsynthesized developed parametric uncertainty system.obtained Application of results, a developed method is the provided; synthesized method, were comparedof toa results, withexample previously developed methods. system was examined via simulation modeling. Synthesis obtained with the help of system was examined via simulation modeling. Synthesis results, obtained with the help of developed developed method, were compared to results, obtained with previously developed methods. system was examined via simulation modeling. Synthesis results, obtained with the help of developed method, were compared to results, results, obtained with previously developed methods. method, were compared to obtained with previously developed methods. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: robust control, adaptive control, parametric uncertainty, aperiodic transient process, control method, were compared to results, obtained with previously developed methods. Keywords: robustsimulation control, adaptive control, parametric uncertainty, aperiodic transient process, control system synthesis, modeling. Keywords: robust control, control, Keywords: robustsimulation control, adaptive adaptive control, parametric parametric uncertainty, uncertainty, aperiodic aperiodic transient transient process, process, control control system synthesis, modeling. Keywords: robust control, adaptive control, parametric uncertainty, aperiodic transient process, control system system synthesis, synthesis, simulation simulation modeling. modeling. system synthesis, simulation modeling. required data, simple structure of a controller, capability to 1. INTRODUCTION required data, simple structurevariations of a controller, to compensate rapid parametric in widecapability ranges and 1. INTRODUCTION required simple structure of capability to required data, data, simple structurevariations of aa controller, controller, capability to compensate rapid parametric in wide ranges and 1. INTRODUCTION required data, simple structure of a controller, capability to 1. INTRODUCTION stability of control quality indices. This is the main aim of the A vast variety of control objects or systems are known to compensate rapid parametric variations in wide ranges and compensate rapid parametric variations in wide ranges and 1. INTRODUCTION stability of control quality indices. This is the main aim of the A vast variety of control objects or systems are known to compensate rapid parametric variations in wide ranges and paper: to develop a method of controller synthesis with have parametric uncertainty. Such uncertainty must be taken stability of of control control quality quality indices. indices. This This is is the the main main aim aim of of the the A vast variety or are to A vast variety of of control control objects objects or systems systems must are known known to stability paper: toof develop a method of This controller synthesis have parametric Such uncertainty be taken control quality indices. is thethe main ofwith the aforementioned features. In order to reach aim,aim a set of A vast variety ofuncertainty. control or of systems aresystems known to stability into consideration during objects synthesis controlmust for paper: to develop a method of controller synthesis with paper: to develop a method of controller synthesis with have parametric uncertainty. Such uncertainty be taken have consideration parametric uncertainty. Such uncertainty must be taken features. In order to reach the aim, a set of into during synthesis of control systems for aforementioned paper: to develop a method of controller synthesis with problems must be solved: a problem of the paper must have parametric uncertainty. Such uncertainty must be taken sophisticated robotics, such as industrial robots, unmanned aforementioned features. features. In In order order to to reach reach the the aim, aim, aa set set of of into consideration during synthesis of systems for into consideration during synthesis of control control systems for aforementioned problems must be solved: a problem of the paper must sophisticated robotics, such as industrial robots, unmanned aforementioned Inoforder to reach aim, a set of formulated,must an features. algorithm designing anthe adaptive-robust into consideration during synthesis of control systems for problems aerial or underwater vehicles, etc. There arerobots, several common be solved: a problem of the paper must problems must be solved: a problem of the paper must sophisticated robotics, such as industrial unmanned sophisticated robotics, such asetc. industrial unmanned formulated,must an be algorithm designing an the adaptive-robust aerial or underwater vehicles, There arerobots, several common controller problems bedeveloped, solved:of a the problem of paper must algorithm must be applied sophisticated robotics, such as industrial robots, unmanned approaches to the synthesis of control systems with uncertain formulated, an algorithm of designing an adaptive-robust formulated,must an be algorithm of the designing an must adaptive-robust aerial or or underwater underwater vehicles, vehicles, etc. etc. There There are are several several common common controller aerial developed, algorithm be system, applied approaches to the synthesis of neural control systems with formulated, anofbe algorithm of athe designing anfor adaptive-robust a problem synthesizing controller some aerial or underwater vehicles, There are several common parameters: adaptive control; networks baseduncertain control; to controller must developed, algorithm must be applied controller must be developed, the algorithm must be system, applied approaches to the synthesis synthesis of etc. control systems with uncertain approaches to the of control systems with uncertain to a problem ofbesynthesizing athecontroller for some parameters: adaptive control; neural networks based control; controller must developed, algorithm must be applied synthesized system must be examined via simulation approaches to the synthesis of control systems with uncertain fuzzy logic based control and robust control, based on to a problem of synthesizing a controller for some system, to a problem system of synthesizing a controller forviasome system, parameters: adaptive adaptive control; control; neural neural networks networks based based control; control; synthesized parameters: must be examined simulation fuzzy logic based control and robust control, based on to a problem of synthesizing a controller for some system, modeling. Let us consider solving these problems further. parameters: control; and neural networks basedbased control; considering system parameters asrobust intervals. Each of these system must be via simulation synthesized mustsolving be examined examined via further. simulation fuzzy logic adaptive based control control control, on synthesized fuzzy logic based andasrobust control, on modeling. Letsystem us consider these problems considering system parameters intervals. Eachbased of these system mustsolving be examined via further. simulation fuzzy logic has based control andasrobust control, based on synthesized approaches its own advantages and disadvantages. For modeling. Let us consider these problems modeling. Let us consider solving these problems further. considering system parameters intervals. Each of these considering system parameters as intervals. Each of these approaches has its own advantages and disadvantages. For modeling. Let us consider solving these problems further. 2. FORMULATION OF A PROBLEM considering system parameters as intervals. Each of these example, adaptive control systems provide stable control approaches has advantages and disadvantages. For approaches has its its own own advantages and disadvantages. For FORMULATION OF A PROBLEM example,butadaptive control systems provide stable control approaches has its compensate own advantages disadvantages. For quality, cannot rapid and parametric variations in Developed 2. 2.method FORMULATION OF an A PROBLEM PROBLEM example, adaptive control systems provide stable control 2. FORMULATION OF A is based on interval approach to example, adaptive control systems provide stable control quality, butadaptive cannot compensate rapid parametric variations in Developed 2.method FORMULATION OF an A PROBLEM example, control provide control wide ranges (Ioannou 2010,systems Ioannou 2012, stable Feng 1999). is based on interval approachof toa quality, but cannot compensate rapid parametric variations in control. According to this approach, each parameter quality, but cannot compensate rapid parametric variations in Developed method is based on approach wide ranges (Ioannou 2010, systems Ioannou 2012, desired Feng 1999). method to is this based on an an interval interval approachof to toa quality, but cannot compensate rapid parametric variations in Developed Neural network based control provide control control. According approach, each parameter wide ranges (Ioannou 2010, Ioannou 2012, Feng 1999). Developed method is based on an interval approach toa control system may vary in some intervals. Each control wide ranges (Ioannou 2010, systems Ioannouprovide 2012, desired Feng control 1999). control. According to this approach, each parameter of Neural network based control control. According to this approach, each parameter of a wide ranges (Ioannou 2010, Ioannou 2012, Feng 1999). quality, but require a vast amount of data to train a neural control system may vary invia some intervals. Each control Neural network network based based control control systems systems provide provide desired desired control control control. According to this approach, each parameter of a Neural system can be described denominator of a transfer quality, but require a control vastAlso, amount ofprovide data to desired train a control neural control system may vary in some intervals. Each control control system may vary in some intervals. Each control Neural network based systems network (Wang 2009). both adaptive and neural system system can be may described viasome denominator of a transfer quality, but but require require aa vast vast amount amount of of data data to to train train aa neural neural control vary in intervals. Each control quality, function – characteristics polynomial. Let us designate network (Wang 2009). adaptive anda neural can be via of aa transfer quality, require a vastAlso, amount of data to train neural system be described described polynomial. via denominator denominator ofdesignate transferaa network but based controllers have aboth complex structure. Fuzzy system network (Wang 2009). Also, both adaptive and neural neural function can – characteristics Let uswith network (Wang 2009). Also, both adaptive and system can be described via denominator of a transfera characteristic polynomial of a system interval network based controllers have aboth complex structure. Fuzzy function – polynomial. Let designate network (Wang 2009). adaptive neural logic based controller has aAlso, simpler structure, but aand procedure function – characteristics characteristics polynomial. Let us uswith designate a network based controllers have aa complex structure. Fuzzy characteristic polynomial of a system interval network based controllers have complex structure. Fuzzy function – characteristics polynomial. Let us designate as interval characteristic polynomial (ICP). Wea logic based controller a have simpler structure, but depends a procedure characteristic polynomial of aa system with interval network based a complex structure. Fuzzy of designing acontrollers rule has base is not strict and on parameters characteristic polynomial of system with interval logic based controller has a simpler structure, but a procedure parameters interval characteristic polynomial Wea logic based controller a simpler structure, but depends a procedure polynomial of aas interval system with(ICP). interval will consideras of ICP parameters of designing rule has base is ofnotan strict and on characteristic parameters as interval polynomial (ICP). We logic based controller has a simpler structure, but(Lee a procedure experience anda qualification engineer 2001). ascoefficients interval characteristic characteristic polynomial (ICP). of Wea of rule and depends on will consider coefficients of ICP as interval parameters of of designing designingandaa qualification rule base base is is ofnot notanstrict strict and (Lee depends on parameters parameters as interval characteristic polynomial (ICP). Wea experience engineer 2001). system, so, general form ofofICP can be writtenparameters as follows:of of designing a rule base is not strict and depends on Procedures of interval control system synthesis are strictly will consider coefficients ICP as interval will consider coefficients of ICP as interval parameters of experience and qualification of an engineer (Lee 2001). experience of andinterval qualification of an engineer (Lee 2001). will system, so, general form ofofICP can be writtenparameters as follows:of aa n coefficients n ICP consider as interval Procedures control system synthesis are strictly experience and qualification of andata: engineer (Lee 2001). formulated, they requirecontrol minimum only minimal and system, general form can i of system, so, general of ICP ICP be written as as follows: follows: Procedures of of interval system synthesis are strictly strictly n q ( K ) form n q can D( s, K )so, ; qi be s ii ,written Procedures interval control system synthesis are i can i si so, general form of ICP be written as follows: (1) formulated, they require minimum data: onlydesired minimal and system, n n Procedures of interval control system synthesis are strictly maximal value of each parameter, and provide control n n D ( s, K ) (1) formulated, they they require require minimum minimum data: data: only only minimal minimal and and i n0 q ( K ) i s ii i n0 q i ; q i s ii , formulated, maximal value of each parameter, anddata: provide desired control D q( K ) s i 0 q (1) D(( ss,, K K )) qii ;; q qii ss i ,, (1) formulated, they require minimum only minimal and quality rapid parametric variations in wide ranges, but i 0 q ( K ) ii s i maximaldespite value of each each parameter, and provide provide desired control D ( s , K ) q ( K ) s q ; q s , (1) maximal value of parameter, and desired control i i 0 0 0 0 i i i quality despite rapid parametric variations in wide ranges, but i i D ( s , K ) where – is an ICP, – is a vector of controller maximal value of each parameter, and provide desired control K synthesized system has insufficient dynamics and value of its i 0 i 0 quality despite despite rapid rapid parametric parametric variations variations in in wide wide ranges, ranges, but but quality where D ( s, K ) – is an ICP, K – is a vector of controller synthesized system insufficient dynamics and value ofbut its quality indices despite rapidhas parametric invariations wide ranges, may vary during variations parametric (Polyak where aa vector controller synthesized system has insufficient dynamics and of D (( ss,, K K )) – where D – )is is an anisICP, ICP, – is is vector of of controller K synthesized system has insufficient dynamics and value value of its its parameters, an K ICP– coefficient function of quality indices may vary during parametric variations (Polyak where D ( s, Kqq i)(K – )is – an ICP, – is a vector of controller K synthesized system has insufficient dynamics and value of its 2002, Bhattacharyya 1995). So, parametric a problem of developing new parameters, quality indices may vary during variations (Polyak (K – is an ICP coefficient function of quality indices may vary during parametric variations (Polyak i 2002, Bhattacharyya 1995). So, parametric a for problem of developing new parameters, )) – coefficient function of quality indices may vary during variations (Polyak methods of controller synthesis systems with uncertain (K – is is qan ani , qICP ICP coefficient function of parameters, minimal and maximal parameters ofqqaii (K controller, 2002, Bhattacharyya 1995). So, aa problem of developing new i – are 2002, Bhattacharyya 1995). So, problem of developing new q (K ) – is an ICP coefficient function of parameters, methods ofiscontroller synthesis for systems with uncertain parameters of ai controller, q i , q i – are minimal and maximal 2002, Bhattacharyya 1995). So, a problem of developing new parameters highly relevant. methods of controller synthesis for systems with uncertain q , q – are minimal and maximal parameters of a controller, methods ofiscontroller synthesis for systems with uncertain values of each ICP coefficient, parameters of a q minimal and maximal parameters of a controller, i , qwhich i – areinclude parameters highly relevant. i i methods ofiscontroller synthesis for systems with uncertain values q i , qwhich minimal and maximal parameters of aICP controller, of each coefficient, parameters of a i – areinclude parameters relevant. parameters is highly highly relevant. One of ways to improve existing method is to combine it controller. values of each ICP coefficient, which include parameters of values of each ICP coefficient, which include parameters of aa parameters is highly relevant. controller. One of ways to improve existing method is to combine it values of each ICP coefficient, which include parameters of a with the other one. This paper is dedicated to a development controller. controller. One of of ways ways to to improve improve existing existing method method is is to to combine combine it it Roots One of (1) are located in some areas of a complex plane due with the other one. This paper is dedicated to a development controller. One of ways improve existing method is atodevelopment combine it Roots of (1) are located in some areas of a complex plane due of a combined method controller synthesis, based on an with the other to one. This of paper is dedicated dedicated to with the other one. This paper is to a development ICPof coefficients variations. This causes variation plane of values of a combined method of controller synthesis, based on an to Roots (1) in areas of due with the other one. This paper is dedicated to a development of (1) are are located located in some some areas of aa complex complex plane due adaptive approach and an interval approach simultaneously. to ICP coefficients variations. This causes variationdegree. of values of aa combined combined method method of of controller controller synthesis, synthesis, based based on on an an Roots Roots of (1) are located in some areas of a complex plane due of of quality indices: stability degree and oscillability In adaptive approach and an interval approach simultaneously. to ICP coefficients variations. This causes variation of values ICP coefficients variations. Thisand causes variationdegree. of values of a combined method ofcombine controller synthesis, an to Developed method willan advantages ofbased theseontwo of quality indices: stability degree oscillability In adaptive approach and interval approach simultaneously. to ICP coefficients variations. This causes variation of values adaptive approach and an interval approach simultaneously. order to reduce oscillation of control quality, a poles Developed method willan combine of these two of of quality quality indices: indices: stability stability degree degree and and oscillability oscillability degree. degree. In In adaptive approach and interval advantages approach simultaneously. approaches: sufficient dynamics of a system, minimum order to reduce oscillation of control quality, a poles Developed method will combine advantages of these two quality degree oscillability In Developed will dynamics combine advantages of these two of approaches: method sufficient of a system, minimum order to indices: reduce stability oscillation of and control quality,degree. a poles poles order to reduce oscillation of control quality, a Developed method will combine advantages of these two approaches: approaches: sufficient sufficient dynamics dynamics of of aa system, system, minimum minimum order to reduce oscillation of control quality, a poles approaches: sufficient dynamics Federation of a system, minimum 2405-8963 © 2018, IFAC (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2018 IFAC 826 Peer review©under of International Federation of Automatic Copyright 2018 responsibility IFAC 826Control. Copyright © 826 10.1016/j.ifacol.2018.11.443 Copyright © 2018 2018 IFAC IFAC 826 Copyright © 2018 IFAC 826
IFAC CAO 2018 Yekaterinburg, Russia, October 15-19, 2018Ivan V. Khozhaev / IFAC PapersOnLine 51-32 (2018) 826–831
dominance principle can be applied. According to this principle, in order to provide desired control quality, one or two poles of a system (dominant poles) must be placed on a complex plane near an imaginary axis; all other poles must be placed far enough away of dominant poles to minimize their influence on a transient process. An example of such allocation is shown in the figure 1.
827
through achieving stability of B ( s, K ) , despite its interval parametric uncertainty, and simultaneously.
R ( s, K )
equality to zero
3. DERIVATION OF GENERAL EXPRESSIONS Let us derive expressions, which allow calculating remainder and coefficients of non-dominant polynomial. As far, as we know a general form of ICP (1) and a dominant polynomial (3), we can derive an expression of B ( s, K ) coefficients and a remainder R ( s, K ) . These expressions can be written as follows:
bi q( K )i1 bi1 ; n
j n 1...0,
R( K ) q( K ) i i D( , K ),
(4) (5)
i 0
where
b i
– are interval coefficients of non-dominant
polynomial B ( s, K ) ; q( K ) i – are interval coefficients of ICP, which depend on parameters of a controller; – is a value of a dominant pole, which defines control quality of a
Fig. 1. Example of poles allocation of a system with interval parameters
system; R(K ) – is a remainder of ICP D( s, K ) division by dominant polynomial A(s ) . Let us formulate a synthesis algorithm on a base of (4) and (5).
Allocation of system poles, given in the figure 1, provides an aperiodic transient process, which duration is determined by a dominant pole, which lies in an interval 6;5 . This causes a variation of stability degree in an interval 5;6 . Such allocation can be provided with the help of several methods (Nesenchuk 2002, Nesenchuk 2008, Nesenchuk 2017, Wang 2002, Zhmud 2014, Zhmud 2017, Tagami 2003, Gayvoronskiy 2014, Gayvoronskiy 2015, Gayvoronskiy 2017). The aim of the paper is to improve these methods, by placing a dominant pole in a point of a complex plane, but not in an interval. This will improve stability of quality indices of a system.
4. DEVELOPMENT OF THE SYNTHESIS ALGORITHM In order to formulate the synthesis algorithm, we will define required data, algorithm itself and its resulting data. First of all, a desired control quality must be defined. The method is based on a pole dominance principle, so desired control quality is defined through defining location of dominant and non-dominant poles. Location of dominant pole corresponds to desire stability degree of a system. Location of non-dominant poles corresponds to desired oscillability degree and distance to dominant pole.
Developed method is based on decomposing ICP in three parts: a dominant polynomial A(s ) ; a non-dominant polynomial B ( s, K ) and a remainder R ( s, K ) : D( s, K ) A( s) B( s, K ) R( K ).
(2)
Desired control quality can be provided in three steps: defining A(s ) according to desired control quality; providing a stability of B ( s, K ) and providing equality of R ( s, K ) to zero. The paper is dedicated to a development of a method of providing an aperiodic transient process, so we should place one real dominant pole according to desired value of stability degree: A( s ) s ,
(3)
where – is a dominant pole. Now we can finally formulate the problem of the research: to develop a method of synthesizing parameters of a controller, which will provide a dominant pole allocation in a point of a complex plane
Fig. 2. Example of defining desired control quality of the system through placing its poles on a complex plane
827
IFAC CAO 2018 828 Yekaterinburg, Russia, October 15-19, 2018Ivan V. Khozhaev / IFAC PapersOnLine 51-32 (2018) 826–831
In the figure 2, stability degree of the system is determined by dominant pole ; robust oscillability degree is determined by sector angle of non-dominant poles allocation; invariance of control quality from non-dominant poles is provided by their maximal real part .
where – is a maximal real part of a non-dominant pole. In order to place non-dominant poles in a sector, a following substitution must be performed:
After the aim of synthesis is set, a controller structure must be chosen. The method requires a controller with at least two tunable parameters: one for stabilizing a dominant pole in a point of a complex plane and the other for placing nondominant poles in desired area of a complex plane. Author recommends using a PI-controller, if there are no special requirements to systems oscillability and it is enough to provide some minimal real part of non-dominant poles. In the other case, if non-dominant poles must be placed in a sector, as it is shown in the figure 2, then a PID-controller should be used. Using a PID-controller allows using two tunable parameters for non-dominant poles placing and makes this problem easier. PID-controller with a following transfer function will be used in all examples of the method application in the paper:
where – is a desired maximal oscillability degree of a system. Constant values of the rest of controller coefficients, chosen from a D-partition, will provide desired allocation of non-dominant poles.
s
(9)
Summarizing all aforementioned information, a synthesis algorithm can be formulated:
KI (6) , s where K P , K I – are proportional and integral coefficients of the PI-controller. Once a controller is chosen, a transfer function of the system with interval parameters can be derived. Then, its denominator – an ICP of a system in a form (1) – can be used
1. Choose desired control quality by defining stability degree and oscillability degree of the system; define a dominant pole and parameters of non-dominant poles allocation area according to a desired control quality. 2. Choose an appropriate controller; derive a transfer function of the system and its ICP in a form (1). 3. Find a remainder R(K ) with the help of (5); solve an equation R( K ) 0 and find a proportional coefficient function of other controller parameters (7).
in (5) to derive a remainders R(K ) function of controller parameters and interval parameters of a system. By equating
4. Substitute (7) to (1) and calculate coefficients of B ( s, K ) with the help of (4).
remainder R(K ) to zero and expressing a proportional coefficient of a controller in terms of other parameters of a system, a law of its adaptation to parametric uncertainty of a system can be derived. General form of this law can be written as follows in case of using a PI-controller (6): (7)
5. Substitute (8) or (9) to B ( s, K ) and derive equations of Dpartition curves; choose values of controller parameters from D-partition and substitute them to (7). 6. Finally, a controller is synthesized; it consists of an adaptation law for a proportional coefficient and constant values of all other parameters.
where q – is a vector of interval parameters of a system. After derivation of (7), it must be substituted to (1). Since coefficients of an ICP depend on one (in case of using a PIcontroller) or two (in case of using a PID-controller),
Let us consider an example of controller synthesis with the help of the method.
coefficients of non-dominant polynomial B ( s, K ) can be calculated via (4). In order to provide desired allocation of
5. EXAMPLE OF SYNTHESIS METHOD APPLICATION Let us assume that there is a problem of providing a desired control quality in a control system with interval parametric uncertainty. Transfer function of a control object of the system is given below:
non-dominant poles, which are roots of B ( s, K ) , a Dpartition method can be used. An equation of a D-partition curve can be derived by substituting s in B ( s, K ) with a border equation of a non-dominant poles allocation area. For example, in order to provide a minimal real part of nondominant poles, following substitution must be performed: s j , 0,
j , 0,
Final form of a controller, synthesized with the help of the method, consists in a law of adaptation of a proportional coefficient, which stabilizes a dominant pole in a specified point of a complex plane, and constant values of other parameters of a controller, which provide a desired allocation of non-dominant poles.
WPI (s, K P , K I ) K P
K P f ( K I , q),
1
W (s, p, q)
(8)
828
[q1 ] s [q0 ] , [ p2 ] s 2 [ p1 ] s [ p0 ]
[q1 ] [7;10], [q0 ] [15;20], [ p 0 ] 30;40, where [ p1 ] [12;15], [ p 2 ] [5;8] – are interval parameters of the system. According to synthesis algorithm, we will define a
IFAC CAO 2018 Yekaterinburg, Russia, October 15-19, 2018Ivan V. Khozhaev / IFAC PapersOnLine 51-32 (2018) 826–831
desired stability degree and define a dominant pole (1; j 0) . Also a maximal real part of non-dominant poles is defined as follows: 2 .
K P ( p, q) 16
([ p 0 ] K P [q 0 ] K I [q1 ]) s K I [q 0 ].
p0 p1 p2 ; q0 q1
K I 16 .
Besides a control object, the system has a PI-controller with a (6) transfer function. Serial connection of the control object and the controller is enclosed with a unity feedback. ICP of the system is given below: D( s, p, q, K P , K I ) [ p 2 ] s 3 ([ p1 ] K P [q1 ]) s 2
829
Poles allocation of a synthesized system is shown in the figure 3. Figure 3 shows, that the synthesis was successful: a dominant pole is stabilized in point (1; j 0) of a complex plane; all other poles are located on the left of a line Im( x) 2 .
(10)
After a transfer function is derived, its denominator can be used to derive a remainder function with the help of (5):
R( p, q, K P , K I ) [ p 2 ] 3 ([ p1 ] K P [q1 ]) 2 ([ p0 ] K P [q0 ] K I [q1 ]) K I [q0 ] [ p 2 ] [ p1 ] K P [q1 ] [ p0 ] K P [q0 ] K I [q1 ] K I [q0 ]. Dominant pole will be placed as defined, if remainder is equal to zero. Let us derive and solve this equation: R ( p, q, K P , K I ) 0; K P ( p, q, K I ) K I
p0 p1 p 2 . q0 q1
(11)
A law of adaptation for proportional coefficient of the PIcontroller (10), which will allow stabilizing a dominant pole in a point of a complex plane, is now obtained. According to a synthesis algorithm, (11) must be substituted to (10). Considering this, an ICP of the system can be written as follows: Fig. 3. Allocation of poles of the synthesized system on a complex plane
3
D( s, p, q, K I ) [5.0;8.0] s 7.0 K I 38 .400 ;10 .0 K I 0.385 s 2 22 .0 K I 78 .0;30 .0 K I 9.231 s [15 .0 K I ;20 .0 K I ].
6. SIMULATION MODELING OF A SYNTHESIZED SYSTEM
Coefficients of this form of ICP can be used to calculate
In order to examine effectiveness of the synthesized controller, simulation modeling should be performed. A model of the system was developed with the help of MATLAB software. The model is shown in the figure 4. The model includes several generators, which provide values of interval parameters of the system, joined in the Interval Parameters subsystem; a block of transfer function with variable parameters, which represent the Control Object of the system and an adaptive PI-controller, synthesized in a previous paragraph.
coefficients of non-dominant polynomial B ( s, K ) with the help of (4):
B( s, K I ) [b2 ] s 2 [b1 ] s [b0 ]; [b2 ] [5.0;8.0]; [b1 ] 7.0 K I 46.400;10.0 K I 5.385 ;
(12)
[b0 ] 3.0 K I 33.015;3.0 K I 46.015 . Equations of D-partition can be derived by performing a substitution (8) and expressing an integral coefficient K I of the controller. Intersection of stability domains, found from D-partition in each vertex of coefficients polytope of (12), provides us with constant value of K I , which allocates nondominant poles as necessary. Let us choose K I 16 and substitute it to (11). Finally, a PI-controller, which provides desired poles allocation despite parametric uncertainty of the system, can be described as follows:
Fig. 4. Matlab model of the synthesized system
829
IFAC CAO 2018 830 Yekaterinburg, Russia, October 15-19, 2018Ivan V. Khozhaev / IFAC PapersOnLine 51-32 (2018) 826–831
Values of interval parameters vary in previously defined intervals with various frequencies as shown in the figure 5. Frequencies are chosen to provide all combinations of values during a period of q1 oscillation.
in following parameters: proportional coefficient K P 2 , integral coefficient K I 2.558 and differential coefficient K D 0.123 . Step responses of two synthesized systems are shown in the figure 7.
Fig. 5. Values of interval parameters of the system
Fig. 7. Comparison of step responses of systems with an adaptive robust controller and robust controller with constant parameters
In order to perform simulation modeling of the system, a form of input signal must be chosen. Most of control systems with uncertain parameters are synthesized with an assumption that parameters of a system vary much slower, than input signal of the system. In order to examine synthesized system in more rigid conditions, we will use a sine function, which varies slower, than parameters of the system, as an input signal. Reaction of the system on such input signal is shown in the figure 6.
Figure 7 shows, that adaptive robust PI-controller is far more effective, than a robust PID-controller with constant parameters. Mean deviation between input and output signals of the system is equal to 0.967% for adaptive robust PIcontroller, synthesized with the help of proposed method, and 3.618% for robust PID-controller with constant parameters. 7. CONCLUSIONS The main aim of the research was reached: a new method of synthesis, which allows designing adaptive robust controllers, was developed. Controllers, synthesized with the help of the method, have advantages of interval and adaptive approaches to manipulating systems with uncertain parameters: a procedure of synthesis is simple; a controller structure is simple and strictly defined; a controller is able to damp rapid significant parametric disturbances and provides stable control quality. All these features were proved by simulation modelling and comparison with a controller, synthesized with the help of another method (Gayvoronskiy 2017). ACKNOWLEDGMENT The reported study is supported by the Ministry of Education and Science of Russian Federation (project #2.3649.2017/PCh).
Fig. 6. Sine input and output of the synthesized system
REFERENCES
Figure 6 shows, that the controller, synthesized with the help of the developed method, is effective in conditions of interval parametric uncertainty.
Ioannou, P., Baldi, S. Robust Adaptive Control. (2010). Levine S. W. (ed.), The Control Systems Handbook, Second Edition: Control System Advanced Methods, 35. CRC Press, Boca Raton, Florida. Ioannou, P.A., Sun, J. (2012). Robust Adaptive Control, p. 821. Courier Corporation, United States.
Let us synthesize a robust PID-controller with constant parameters for the same system with the help of method, described in (Gayvoronskiy 2017), and compare results of simulation modeling. Synthesis of the PID-controller resulted 830
IFAC CAO 2018 Yekaterinburg, Russia, October 15-19, 2018Ivan V. Khozhaev / IFAC PapersOnLine 51-32 (2018) 826–831
Feng, G., Lozano, R. (1999). Adaptive Control Systems, p. 352. Newnes, Oxford, United Kingdom. Wang, D., Huang, J., Lan, W., Li, X. (2009). Neural networkbased robust adaptive control of nonlinear systems with unmodeled dynamics. Mathematics and Computers in Simulation 79(5), pp. 1745-1753. Lee, H.J., Park, J.B., Chen, G. (2001). Robust fuzzy control of nonlinear systems with parametric uncertainties. IEEE Transactions on Fuzzy Systems, 9, № 2, pp. 369-379. Polyak, B.T, Scherbakov, P.S. (2002). Robust stability and control, p.303. Science, Moscow. Bhattacharyya, S.P, Chapellat, H., Keel L.H. (1995). Robust control: The parametric approach, p. 672. Prentice-Hall, United States Nesenchuk, A.A. (2002). Parametric synthesis of qualitative robust control systems using root locus fields. Proceedings of the 15th Triennial World Congress of The International Federation of Automatic Control (IFAC) Barcelona, Spain 21–26 July 2002. pp. 387-387. Nesenchuk, A.A., Fedorovich, S.M. (2008). Parametric synthesis method of integral systems on the basis of root locus curves of Kharitonov polynomials. Automation and Remote Control, 69 (7), pp. 1133-1141. Nesenchuk A.A. (2017). A method for synthesis of robust interval polynomials using the extended root locus. Proceedings of the American Control Conference 24 -26 May 2017, Sheraton Seattle; United States, pp. 1715172. Wang, Y., Schinkel, M., K.J. Hunt. (2002). PID and PID-like controller design by pole assignment within D-stable regions. Asian Journal of Control, 4, № 4, pp. 423-432. Zhmud, V., Dimitrov, L., Yadrishnikov, O. (2014). Calculation of regulators for the problems of mechatronics by means of the numerical optimization method. Proceedings of 12th International Conference on Actual Problems of Electronic Instrument Engineering, APEIE 2014; Novosibirsk; Russian Federation; 2 - 4 October 2014, pp. 739-744. Zhmud, V.A., Reva, I.L., Dimitrov, L.V. (2017). Design of robust systems by means of the numerical optimization with harmonic changing of the model parameters. Journal of Physics: Conference Series, 803 (1), article number 012185. Tagami, T. Ikeda, K. (2003). Design of robust pole assignment based on Pareto-optimal solutions. Asian Journal of Control, 5(2), pp. 195-205. Gayvoronskiy S.A., Ezangina T. (2014). The synthesis of the robust stabilization system of cable tension for the test bench of weightlessness simulation. Advanced Materials Research, 1016, pp. 394-399. Gayvoronskiy S.A., Ezangina T., Khozhaev I. (2015). The interval-parametric synthesis of a linear controller of speed control system of a descent submersible vehicle. IOP Conference Series: Materials Science and Engineering, 93, article number 012055. Gayvoronskiy S.A., Ezangina T., Khozhaev I. (2017). Method of interval system poles allocation based on a domination principle. International Conference on Mechanical, System and Control Engineering, ICMSC
831
2017; St. Petersburg; Russian Federation; 19 May 2017 21 May 2017, pp. 245-249.
831