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Teaching multiloop control of nonlinear system: three tanks case study
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Teaching multiloop control of nonlinear system: three tanks case study Teaching multiloop control of nonlinear system: three tanks case study Teaching multiloop control of nonlinear system: three tanks case study Teaching multiloop control of nonlinear system: three tanks case study * *
Danica Rosinová*, Peter Balko*, Teofana Puleva ** Danica Rosinová**, Peter Balko**, Teofana Puleva ** Danica Peter Balko Balko ,, Teofana Teofana Puleva Puleva ** ** Danica Rosinová Rosinová ,, Peter * Institute of Automotive Mechatronics Slovak University of Technology in Bratislava, Slovakia (e-mail: * Institute of Automotive Mechatronics Slovak University of Technology in Bratislava, Slovakia (e-mail: * Slovak of [email protected]; [email protected]). * Institute Institute of of Automotive Automotive Mechatronics Mechatronics Slovak University University of Technology Technology in in Bratislava, Bratislava, Slovakia Slovakia (e-mail: (e-mail: [email protected]; [email protected]). [email protected]; [email protected]). ** Faculty of Automatics, Technical University of Sofia, Sofia, [email protected]; [email protected]). ** Faculty of Automatics, Technical University of Sofia, Sofia, ** of Technical University Bulgaria; (e-mail: [email protected]). ** Faculty FacultyBulgaria; of Automatics, Automatics, Technical University of of Sofia, Sofia, Sofia, Sofia, (e-mail: [email protected]). Bulgaria; Bulgaria; (e-mail: (e-mail: [email protected]). [email protected]). Abstract: The paper focuses on teaching advanced MIMO control topics using hydraulic system model. Abstract: The paper focuses on teaching advanced MIMO control topics using hydraulic system model. Abstract: focuses on teaching teaching MIMOare control topics using tank hydraulic Basic stepsThe of paper nonlinear MIMO system advanced control design shown on three plantsystem model.model. After Abstract: focuses on MIMO control topics using hydraulic Basic stepsThe of paper nonlinear MIMO system advanced control design are shown on three tank plantsystem model.model. After Basic steps of nonlinear MIMO system control design on three model. After modelling part, comprising nonlinear model based on are firstshown principles andtank its plant linearization about Basic steps of nonlinear MIMO system control design are shown on three tank plant model. After modelling part, comprising nonlinear model based on first principles and its linearization about modelling nonlinear model based on principles linearization about equilibriumpart, point,comprising two approaches to MIMO system control are demonstrated. Theitsfirst approach designs modelling nonlinear model based on first first principles and and linearization about equilibriumpart, point,comprising two approaches to MIMO system control are demonstrated. Theitsfirst approach designs equilibrium point, two approaches to MIMO system control are demonstrated. The first approach designs aequilibrium full state feedback PI controller in state space, the second one uses multi-loop approach with individual point, two approaches to MIMO system control are demonstrated. The first approach designs a full state feedback PI controller in state space, the second one uses multi-loop approach with individual aaloop full state PI in space, second one multi-loop approach individual design, overall stability condition is used tothe detune loop PIuses controllers for the overallwith system issues. fulldesign, state feedback feedback PI controller controller in state state space,to the second one multi-loop approach individual loop overall stability condition is used detune loop PIuses controllers for the overallwith system issues. loop design, overall stability condition is used to detune loop PI controllers for the overall system issues. The results are analysed and compared, important teaching issues are indicated. loop design,are overall stability condition isimportant used to detune loop PI controllers for the overall system issues. The results analysed and compared, teaching issues are indicated. The results are analysed and compared, important teaching issues are indicated. The results are analysed and compared, important teaching issues are indicated. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier rights reserved. Keywords: modelling, multi-input multi-output system, multi-loop control, Ltd. PI All controller, nonlinear Keywords: modelling, multi-input multi-output system, multi-loop control, PI controller, nonlinear Keywords: modelling, multi-input multi-output system, multi-loop control, PI controller, nonlinear system, robust stability Keywords: modelling, system, robust stability multi-input multi-output system, multi-loop control, PI controller, nonlinear system, system, robust robust stability stability 1. INTRODUCTION reliability improvement. Robustness is one of attractive 1. INTRODUCTION reliability improvement. Robustness is one of attractive 1. INTRODUCTION improvement. Robustness is one attractive qualities a multi-loop control scheme, such control Teaching advanced 1.control systems belongs to important reliability INTRODUCTION reliabilityof Robustness is since one of of attractive qualities ofimprovement. a multi-loop control scheme, since such control Teaching advanced control systems belongs to important structure qualities of a multi-loop control scheme, since such control can be inherently resistant to a wide range of Teaching advanced control systems belongs to important topics in mechatronics study. To connect theoretical concepts qualities of a multi-loop control scheme, since such control Teaching advanced control systems belongs to important structure can be inherently resistant to a wide range of topics in mechatronics study. To connect theoretical concepts structure can be inherently resistant to a wide range of uncertainties both in subsystems and interconnections. topics in mechatronics study. To connect theoretical concepts with applications, modelling and simulation of real plants structure can be inherently resistant to a wide range of topics in mechatronics study. To connect theoretical concepts uncertainties both in subsystems and interconnections. with applications, modelling and simulation of real plants uncertainties both in subsystems and interconnections. with applications, modelling and simulation of real plants contribute to increase interest and understanding of control uncertainties both in subsystems and interconnections. with applications, modelling and simulation of real plants contribute to increase interest and understanding of control We study the basic steps of nonlinear MIMO system control contribute to interest understanding control design fundaments, students to real of laboratory We study the basic steps of nonlinear MIMO system control contribute to increase increasepreparing interest and and understanding control design design fundaments, preparing students to real of laboratory We the of MIMO on three tanksteps process case study with 4system inputscontrol and 3 We study study the basic basic of nonlinear nonlinear MIMO design fundaments, preparing students to real laboratory work and practice, Oggunaike et al. (1994). Hydraulic design on three tanksteps process case study with 4system inputscontrol and 3 design and fundaments, real laboratory work practice, preparing Oggunaikestudents et al. to (1994). Hydraulic outputs. design on three tank process case study with 4 inputs and 3 After the modelling phase, control structure design on three tank process case study with 4 inputs and is work and practice, Oggunaike et al. (1994). Hydraulic systems belong to basic laboratory plants, used for simple as outputs. After the modelling phase, control structure is3 work and practice, Oggunaike al. (1994). systems belong to basic laboratoryetplants, used for Hydraulic simple as outputs. After the modelling phase, control structure is selected; a full size state feedback control is designed using outputs. aAfter the state modelling phase, control structure is systems belong basic plants, for as well as complex task demonstration, full size feedback control is designed using systems belong to to control basic laboratory laboratory plants, used usedRosinová for simple simpleand as selected; well as complex control task demonstration, Rosinová and selected; aa full size state feedback designed using pole placement technique in thecontrol state is space. Presented selected; full size state feedback control is designed using well as complex control task demonstration, Rosinová and Kozáková, (2009). Many control issues, including placement technique in the state space. Presented well as complex control task demonstration, Rosinová and pole Kozáková, (2009). Many control issues, including pole technique in space. Presented alternative – multi-loop requires a choice of pole placement placement technique control, in the the state state space. Presented Kozáková, (2009). Many control issues, nonlinearities and multi-input multi-output systemincluding control alternative – multi-loop control, requires a choice of Kozáková, (2009). Many control issues, including nonlinearities and multi-input multi-output system control appropriate alternative – multi-loop control, requires a choice input-output pairing. Further step is independent alternative input-output – multi-looppairing. control, requires a independent choice of of nonlinearities and multi-input multi-output system control can be taught and illustrated on various types of hydraulic appropriate Further step is nonlinearities multi-inputonmulti-output system control appropriate can be taught and and illustrated various types of hydraulic Further is single loopsinput-output design so pairing. that it guarantees of the appropriate input-output pairing. Further step stepstability is independent independent can be taught and illustrated on various types of hydraulic plants, as four tank system introduced by Johansson et al. single loops design so that it guarantees stability of the can be as taught on variousbytypes of hydraulic plants, four and tankillustrated system introduced Johansson et al. single so it of overall systemdesign including interactions. Two stability alternatives of single loops loops so that that it guarantees guarantees of the the plants, four system (1999), Johansson (2000). systemdesign including interactions. Two stability alternatives of plants, as as four tank tank system introduced introduced by by Johansson Johansson et et al. al. overall (1999), Johansson (2000). overall system including interactions. Two alternatives of stability condition for decentralized control structure are overall system including interactions. Two alternatives of (1999), Johansson (2000). stability condition for decentralized control structure are (1999), decentralized control structure are used: onecondition based on for small gain theorem for complementary In this Johansson paper, we(2000). consider a three tank system model stability stability condition for decentralized control structure are one based on small gain theorem for complementary In this paper, we consider a three tank system model used: one on small theorem for sensitivity function onegain for systems no RHP (right In wetheconsider aa three system corresponding laboratory plant tank delivered by model Inteco used: one based based onand small theorem with for complementary complementary In this this paper, paper, to three system sensitivity function and onegain for systems with no RHP (right corresponding towetheconsider laboratory plant tank delivered by model Inteco used: sensitivity function and one for systems with no RHP half plane) zeros. The obtained results are analysed and corresponding to plant delivered by company. Series of laboratory control design necessary for sensitivity and one for systems RHP (right (right corresponding to the the plant tasks delivered by Inteco Inteco plane)function zeros. The obtained resultswith are no analysed and company. Series of laboratory control design tasks necessary for half half plane) zeros. The obtained results are analysed compared. company. Series of control design tasks necessary for nonlinear MIMO controller design is presented. half plane) zeros. The obtained results are analysed and and company. Series system of control design tasks necessary The for compared. nonlinear MIMO system controller design is presented. The compared. nonlinear MIMO design is presented. The system system controlcontroller based on linearization around compared. nonlinear MIMO system controller design is presented. The 2. MODELLING AND CONTROL OF NONLINEAR nonlinear system control based on linearization around 2. MODELLING AND CONTROL OF NONLINEAR nonlinear system control based on linearization around working point is adopted. works only nonlinear system control Though based this on approach linearization around 2. AND CONTROL OF THREE SYSTEM working point is adopted. Though this approach works only 2. MODELLING MODELLING ANDTANK CONTROL OF NONLINEAR NONLINEAR THREE TANK SYSTEM working point is adopted. Though this approach works only within limited operation region, its simplicity and working point is adopted. Though this approach works only THREE TANK SYSTEM within limited operation region, its simplicity and In this section, key pointsTANK of teaching basics of nonlinear THREE SYSTEM within operation region, its simplicity applicability in many real plants attractive and In this section, key points of teaching basics of nonlinear within limited limited operation region,make its it applicability in many real plants make itsimplicity attractive and and In points of nonlinear modeling and key control, three basics tank of model, are In this this section, section, pointsusing of teaching teaching nonlinear applicability in many real plants make it attractive and frequently used in nonlinear systems control. Modelling part and key control, using three basics tank of model, are applicability in inmany real systems plants make it Modelling attractive part and modeling frequently used nonlinear control. modeling and using tank model, are summarized. The control, aim of this part ofthree advanced control course modeling and control, using three tank model, are frequently used in nonlinear systems control. Modelling part comprises nonlinear model and its linearization around summarized. The aim of this part of advanced control course frequently used in nonlinear Modelling part summarized. comprises nonlinear model systems and itscontrol. linearization around The aim of this part of advanced control course is to cover basic steps necessary to design appropriate control summarized. The aim of this part of advanced control course comprises model and itsslightly linearization equilibrium point. (The results differentaround from is to cover basic steps necessary to design appropriate control comprises nonlinear nonlinear model andare linearization equilibrium point. (The results areitsslightly differentaround from is basic necessary design forto acover nonlinear MIMO system,to realized and verifiedcontrol on a basic steps steps necessary design appropriate appropriate equilibrium point. (The results are slightly different from those presented in plant user manual.) The main focus is on is fortoacover nonlinear MIMO system,torealized and verifiedcontrol on a equilibrium point. (The results are slightly different from those presented in plant user manual.) The main focus is on for a nonlinear MIMO system, realized and verified simulation model of a three tank system. for a nonlinear system, realized and verified on on aa those presented plant user manual.) The main focus is on controller designin and comparison of different control design simulation modelMIMO of a three tank system. those presented in plant user manual.) The main focus is on controller design and comparison of different control design simulation model of a three tank system. model of a three tank system. controller design comparison of control approaches. Stateand feedback control with full sizedesign gain simulation part includes: controller design comparison of different different control approaches. Stateandfeedback control with full sizedesign gain Modelling Modelling part includes: approaches. State feedback control with full size gain matrices is confronted with a multi-loop controller (called part includes: approaches. State feedback withcontroller full size(called gain Modelling matrices is confronted with a control multi-loop Modelling of part includes:three tank mathematical model, input nonlinear matrices is with multi-loop controller (called also decentralized controller, some sources). matrices is confronted confronted with aa according multi-loopto (called -- analysis analysis of nonlinear three tank mathematical model, input also decentralized controller, according tocontroller some sources). -- analysis nonlinear mathematical outputof variables andthree theirtank ranges, also according some Multi-loop control controller, reduces multivariable problem and analysis nonlinear mathematical model, model, input input also decentralized decentralized according to tocontrol some sources). sources). outputof variables andthree theirtank ranges, Multi-loop control controller, reduces multivariable control problem and and output variables and their ranges, Multi-loop control reduces multivariable control problem complexity, and enables to employ decentralized control and output variables and their ranges, Multi-loop control reducesto multivariable control problem complexity, and enables employ decentralized control - determination of working area – steady state of nonlinear complexity, enables employ structure withand subsystems having theirdecentralized local control control loops. - determination of working area – steady state of nonlinear complexity, enables to to employ structure withand subsystems having theirdecentralized local control control loops. --system determination of working and control determination of aim working area area – – steady steady state state of of nonlinear nonlinear structure with local control loops. Compared withsubsystems centralized having MIMO their controller systems, multi- system and control aim structure with subsystems having their local control Compared with centralized MIMO controller systems, loops. multi- system and control aim system and control aim Compared with centralized MIMO controller systems, multiloop control structure generally yields certain performance Compared with centralized MIMOyields controller systems, multi- - derivation of linearized model using standard Taylor series loop control structure generally certain performance - derivation of linearized model using standard Taylor series loop control structure yields certain deterioration, however,generally weighted against byperformance important approach loop control structure generally yields certain performance -- derivation deterioration, however, weighted against by important approach derivation of of linearized linearized model model using using standard standard Taylor Taylor series series deterioration, however, weighted against by important benefits, such as design simplicity, hardware, operation and deterioration, however, weighted against by important approach benefits, such as design simplicity, hardware, operation and approach benefits, benefits, such such as as design design simplicity, simplicity, hardware, hardware, operation operation and and
2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 IFAC 360Hosting by Elsevier Ltd. All rights reserved. Copyright 2016 responsibility IFAC 360Control. Peer review©under of International Federation of Automatic Copyright © 2016 IFAC 360 10.1016/j.ifacol.2016.07.204 Copyright © 2016 IFAC 360
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- verification of linearized model: comparison of nonlinear and linear model using the respective Simulink schemes; importance of settling the system into the working point
361
difficulties in achieving the control aim - a high accuracy control of the tank levels.
Controller design part consists of:
6 1
- determination of control structure (full size controller versus multi-loop – decentralized controller)
7
- choosing appropriate input-output pairing for multi-loop controller
2
- loop controller design (PI controllers) 3
- testing (robust) stability condition under decentralized controller design - verifying the designed controllers by simulation in Simulink, comparison of linearized model and nonlinear model results.
5
4
Fig. 1 Physical model of multitank system 3.1 Nonlinear model of three tank system
Presented material can be preferrably used for project oriented teaching, when the students have to complete tasks and write a report, summarizing all procedures and results. The important part of the project is analysis of the obtained results and comparison of the designed controllers and closed loop responses both for original nonlinear and linearized model.
The mathematical model of multi tank system can be derived from the general model of n-tanks. The model of the process dynamics can be obtained by the mass balance equation dV1 dH1 q C1H1 1 wl1H1 1 dH1 dt
dV2 dH 2 C1H1 1 C2 H 2 2 wl1H1 1 wl 2 H 2 2 dH 2 dt
3. THREE TANK SYSTEM MODELLING The model of multi tank system, corresponding to the one, produced by Inteco Co is considered, Fig.1. It comprises three separated tanks placed above each other and interconnected with drain valves (Inteco Co., 2006). The upper tank has a constant cross section, while the others have a variable cross section: trapezoidal middle one and round lower one. These variable cross sections add further nonlinearities in the system, besides physical laws of flow. The water is pumped into the upper tank from the supply tank by a pump driven by a DC motor. The water outflows from the tanks due to gravity through orifices placed in the bottom of tanks. The tank valves act as flow resistors. The area ratio of the valves is controlled and can be used to vary the outflow characteristic. Each tank is equipped with a level sensor based on hydraulic pressure measurement.
(1)
dVn dH n Cn 1H n n11 Cn H n n wl ( n 1) H n n11 wln H n n dH n dt where Vi is the fluid volume of the i-th tank, Ci is the resistance of the i-th output orifice, wli is the resistance of the
i-th disturbance orifice and H i is the level in the i-th tank. For the laminar flows, the outflow rate from a tank can be described by the Bernoulli law with i 1/ 2. For real
configuration, a more general coefficients i are applied. From (1), the nonlinear multi tank system model is obtained dH1 1 1 1 q CH 1 w H 1, (2) dt 1 ( H1 ) 1 ( H1 ) 1 1 1 ( H1 ) l1 1
The standard procedure of the model development is adopted, consisting of the following steps: developing the mathematical model based on the basic physical laws; determination of equilibrium respective to the working point and the respective model linearization; building the simulation model and its verification.
dH 2 1 1 1 C1H1 1 C2 H 2 2 w H 1 , (3) dt 2 ( H 2 ) 2 ( H 2 ) 2 ( H 2 ) l1 1
dH 3 1 1 C H 2 C H 3 3 ( H 3 ) 2 2 3 ( H 3 ) 3 3 dt 1 1 wl 2 H 2 2 w H 3 3 ( H 3 ) 3 ( H 3 ) l 3 3
The nonlinear model is developed in the state space, where liquid levels H 1 , H 2 , H 3 in the tanks are measurable state variables of the system. The controlled inputs are: liquid flow rate q and valves settings C1 , C 2 , C3 . Output variables are the same as state ones, which later simplifies state space controller design. In Fig. 1, the multi tank system components interconnection is shown (Inteco, 2006). In this paper the control strategy based on pump and valves control system is investigated. The nonlinearities caused by shapes of tanks and those introduced by valve flow dynamics cause
1 ( H1 ) aw
2 ( H 2 ) cw
H2
H 2 max
bw
,
(4)
(5)
2 2 3 ( H 3 ) w R (R H3 )
where i ( H i ) is the cross sectional area of i-th tank at the level H i . 361
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As mentioned above, the middle and lower tanks have a variable cross sectional area. We assume for simplicity zero disturbances wl1 wl 2 wl 3 .
respective equilibrium, therefore the initial conditions of used integrators must be appropriately preset. The results of nonlinear an linear model for H1 and H2 are shown in Fig.2, result for H3 is similar (step change from equilibrium was performed in time 500).
3.2 Linearized model of the three tank system The linearized models of the upper, middle and lower tanks are obtained by the Taylor expansion of (2)-(4) around the assumed equilibrium state represented by the defined pump flow rate, valves resistance and the tank water levels denoted as q0 , Ci 0 , i 1,3 and Hi 0 , i 1,3 respectively. The state
Table 1. Geometrical parameters of the tanks Upper tank
vector, the control signal, and the output are:
x H1 H 2 H 3 T , u q C1 C2 C3 T ; y=x. (6) In the equilibrium point, the input flow rate of the i-th tank equals to its output flow rate
10 20 q0 C C C30 H3030 . 10 H10 20 H 20
(7)
The state space description of the system linearized around the defined equilibrium point is
x Ax Bu
, y Cx where the components of the state matrix A are given by a Jacobian matrix respective to (2)-(4): C H 1 1 a11 10 1 10 , a12 a13 0 , 1 ( H10 )
H1max 0.35 a = 0.25 m w= 0.035 m
Pump 1.4.104
(8)
Table 3. Level references
linearized model nonlinear model 0.135
0.13
0.125
32 ( H 30 )
0.12 400
.
800
1000
t
1200
1400
1600
0.12
0.115
3
H 30 H 20 , b34 , 3 ( H 30 ) 3 ( H 30 )
H2
linearized model nonlinear model 0.11
1 2 H10 H 20 , b23 , 2 ( H 20 ) 2 ( H 20 ) 2
600
0.125
1 H10 1 , b12 , b b 0, 1 ( H10 ) 13 14 1 ( H10 )
b31 b32 0 , b33
Lower tank 0.1 m
0.14
22 ( H 20 )
3 3 1 2 ' 20 H 20 C30 H 30 3 ( H 30 ) C30 3 H 30 3 ( H 30 )
b21 b24 0 , b22
Middle tank 0.1 m
0.15
The elements of the matrix B(3 x 4) are
b11
Lower 5.5.105
0.12 m
C H 2 1 a31 0 , a32 20 2 20 , 3 ( H 30 )
a33
Middle 5.5.105
0.145
Upper 5.5.105
Upper tank
1 2 2 1 ' 10H10 C20 H 20 2 ( H 20 ) C20 2 H 20 2 ( H 20 )
C
H3max 0.35 m R = 0.364 m w= 0.035 m
3 Valves [m / s]
[ m3 / s ]
H1
a22
Lower tank
Table 2. Flow rate limits
C H 1 1 a21 10 1 10 , a23 0 , 2 ( H 20 )
C
Middle tank H 2max 0.35 m b = 0.345 m c = 0.1 w= 0.035 m
0.105
0.1 400
(9)
and C is identity matrix I (3 x3) .
600
800
1000
t
1200
1400
1600
Fig.2 Model responses for linearized and nonlinear model 4. CONTROLLER DESIGN The main aim of this part is to confront two basic possibilities for MIMO system controller design based on material taught in control courses. In both cases, PI controllers are designed to remove steady state control error.
In Tables 1 and 2, the geometrical parameters and technological limits of the control signals are given. The level references are shown in Table 3, the corresponding steady state pump flow rate is q0 2,4.105 m3 / s .
a) Full MIMO controller designed in state space by pole placement method (Slavov and Puleva (2012))
The linearized model is verified and compared with the nonlinear one in Simulink environment. It is important to underline that verification must be performed around the
b) Multiloop (decentralized) controller. 362
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4.1 Full state feedback controller
0.16 0.14
The first option uses Matlab function place for an augmented system (10), including the additional integral states.
0.12
linear system nonlinear system
(10)
H1
0.1
B A 0 33 x x u 0 34 C 0 33 y C 0 33 x
363
0.08 0.06
0.04
where x xT xiT denotes the augmented state.
0.02
The resulting controller can be divided into its proportional part (feedback gain) and integral part placed in the forward path, see Fig.3.
0
0
500
t
1000
1500
0.14
1 s
0.12
Integrator1
Integrator2 1 s Integrator3
K*u Ki
0.1
x' = Ax+Bu y = Cx+Du
output
State-Space1
To Workspace4
linear system nonlinear system
H2
1 s
K*u
0.08
0.06
Kp 0.04
0.02
Fig. 3. Closed loop with full state feedback PI controller To ensure tank system step response without overshoot and a settling time (up to 200s), the desired poles are chosen as p -0.0333 -0.0367 -0.0400 -0.0433 -0.0467 -0.0500 The obtained controller matrices are 0.1602 0.0795 0.0359 -0.0378 0.1655 0.0650 4 Ki 10 , -0.0222 -0.0920 0.0733 -0.0225 -0.0928 -0.1470 0.7058 0.3604 0.1803 -0.0652 0.7006 0.3264 3 10 . K -0.1424 -0.3184 0.3681 -0.1441 -0.4748 -0.7386 Simulation results for both linear and nonlinear systems with the designed full MIMO PI controller are shown in Fig. 4. Several observations can be made by studying the responses of linear and nonlinear model. These observations belong to crucial student tasks in project completing. While the responses of linearized model does not depend on the working point, the nonlinear model have different behaviour in different operating ranges. The first part of responses up to 500s belong to transient into equilibrium point respective to linearization. After reaching the operating point corresponding to the reference levels in all tanks, listed in Table 3, step responses of the designed control loops are studied: 10% step changes of reference levels are made in 500s for H1, in 700s for H2, and, finally, in 900s for H3. It should be noted that in the working range (after the reference levels were reached), the linearized system dynamics is closer to the nonlinear one, than within transient time up to 500s.
0
500
t
1000
1500
0.13 0.12 0.11 0.1 linear system nonlinear system
H3
0.09 0.08 0.07 0.06 0.05 0.04 0.03
0
500
t
1000
1500
Fig. 4. Comparison of closed loop step responses for a linearized and nonlinear three tank model with full state feedback PI controller 4.2 Multi-loop controller The second controller design approach addresses the multiloop controller and is carried out in frequency domain. The transfer function matrix G(s), corresponding to the linearized model (8), (9) is obtained by standard Matlab conversion. To start a multi-loop controller design, it is inevitable to appropriately choose input-output pairing. In the above three tank model, 4 inputs and 3 outputs are considered. The first task is then to choose 3 of the existing inputs appropriate for a multi-loop design. This can be done analysing the steady state gain matrix. The appropriate choice is pairing input flow q0 to output H1, input valve resistance C20 to output H2, and input valve resistance C30 to output H3, thus omitting the 363
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second input C10. Then, the multiloop controller design is performed for a square transfer function matrix G(s). 114.3 0 0 s 0.0114 45.5 1.644 G( s ) 2 0 s 0.0287 s 0.0002 s 0.0173 36.05 0.0225 36.05s 3 s 0.0424s 2 0.0006s s 2 0.03 s 0.0002 s 0.0137
low frequencies, where TD 1 , for stable system with no
(10)
(11)
E ( G GD )G 1 GM G 1 , SD ( I GD R )1 . The above condition can be reformulated as:
G 1 S D M 0
To find R( s ) , the so called independent design is considered, where individual loops are designed “independently” (simultaneously). Local controllers Ri ( s ) are designed so that they:
(15)
The multiloop controller was designed independently for diagonal transfer functions from (10), resulting controllers were obtained after detuning to satisfy stability condition (15), see Fig. 5. Stability condition (14) is not appropriate since its module is significantly below 1 for low frequencies. Loop controllers for (11) are
b) satisfy the overall system stability condition. In the following, sensitivity is denoted as 1 S( s ) ( I G( s )C( s )) and closed loop transfer function (complementary sensitivity) is denoted as T ( s ) G( s )C( s )( I G( s )C( s )) 1 .
R1( s )
Let us turn to stability condition for a system with multiloop controller. Matrix G(s) can be splitted into its diagonal and off-diagonal parts: G( s ) GD ( s ) GM ( s ) .
8.75s 0.1 4 44s 0.75 4 .10 , R2 ( s ) .10 , s s
R3 ( s )
41.6s 0.57 .10 4 s
1.4
The potential uncertainties (due to linearization) can be included into GM ( s ) . For a stable open loop system G(s)R(s), the closed loop system stability condition based on small gain theorem is given in Veselý and Harsányi (2008).
1.2 1 0.8
Lemma 1
0.6
Consider stable system G(s) with a decentralized controller C(s). The respective closed loop system T(s) is stable if
0.4
abs(S1) abs(S2) abs(S3) 1/norm(Gm*G)
0.2
GD1W GM 1
(12)
or
1 GM
(13)
where matrix W is given by R 1 GD GDW 1 . Inequality (13) can be reformulated into
GD1TD M 0
1 . GM
Either of alternatives (14) or (15) must be satisfied for all frequencies.
a) stabilize individual loops
GD1W
Lemma 2 (Skogestad and Postlethwaite, 2009) Consider stable system G(s) with decentralized controller R(s). Assuming that neither G nor GD has RHP zeros, the 1 overall closed loop system is stable if and only if I ES D is stable, where
A multi-loop PI controller is considered: 1 0 0 P1 T1s 0 0 R1( s ) 1 R( s ) 0 R2 ( s ) P2 0 0 0 T2 s R3 ( s ) 0 0 1 0 P3 0 T3 s
RHP zeros this may be too restrictive. The alternative condition for this case is in Lemma 2.
1 GM
(14)
where TD GD R( I GD R )1 . Condition (14) can be used for stable system without or with RHP zeros (both for minimum and non-minimum phase case). However, the above condition can be rather limiting in 364
0 -4 10
(6a) -3
10
-2
10 frequency
-1
10
0
10
Fig. 5. Overall system stability condition for a multiloop controller. Red line denotes the upper limit to keep stability. Significant point of controller design task is a comparison of responses obtained for full MIMO controller from Section 4.1and responses of multi-loop controller. Step responses in Fig. 6 indicate that multiloop controller outperforms the former one, however, is more sensitive. Control inputs range is similar, though, in multiloop case, rapid changes of control variables appear, Fig. 7. Detail performance indices summary is omitted due to space reasons.
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6
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4 full MIMO controller multiloop controller
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-2 u
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Fig. 7 Control input responses for a full MIMO controller (figure above) and multiloop controller (figure below)
full MIMO controller multiloop controller
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The work has been supported by Grant No 030STU-4/2015 of the Cultural and Educational Grant Agency of the Ministry of Education, Science, Research and Sport of the Slovak Republic.
0.07 0.06 0.05 0.04 0.03
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REFERENCES Johansson, K. H., Horch, A., Wijk, O. and Hansson, A. (1999). Teaching Multivariable Control Using the Quadruple-Tank Process. In: Proc. 38nd IEEE CDC, Phoenix, AZ Johansson, K.H. (2000). The Quadruple-Tank Process: A Multivariable Laboratory Process with an Adjustable Zero. IEEE Transactions on Control Systems Technology, Vol. 8, No. 3, pp.456-465. Multitank system, User’s Manual, Inteco Co, 2006. Ogunnaike, Babatunde A and Harmon W Ray (1994). Process dynamics, modeling, and control. New York: Oxford University Press. ISBN 01-950-9119-1. Rosinová, D. and Kozáková, A. (2009). Robust decentralized PID controller design: a case study. In: IEEE Sankt Peterburg, Russia Skogestad, S. and Postlethwaite, I. (2009). Multivariable feedback control: analysis and design. John Wiley & Sons Ltd., Chichester, West Sussex, UK. Veselý, V. and Haršányi, L. (2008). Robust control of dynamic systems. Vydavateľstvo STU, Bratislava, SR.
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Fig. 6. Step responses of output tank levels for a full MIMO controller and multiloop controller 5. CONCLUSION Control design for a three tank multi-input multi-output system is analysed in this paper from a teaching point of view. The adopted control design approach is based on linearized model of originally nonlinear system around the working point. Steps of the overall control design procedure are indicated including modelling, the corresponding student tasks are proposed. Two approaches: centralized and (8) decentralized control, are compared. Skills gained using control design based on linearized model are very useful also for further studies of LPV (Linear Parameter Varying), gain scheduling and other, more general control schemes for nonlinear systems.
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Teaching multiloop control of nonlinear system: three tanks case study Teaching multiloop control of nonlinear system: three tanks case study Teaching multiloop control of nonlinear system: three tanks case study Teaching multiloop control of nonlinear system: three tanks case study * *
Danica Rosinová*, Peter Balko*, Teofana Puleva ** Danica Rosinová**, Peter Balko**, Teofana Puleva ** Danica Peter Balko Balko ,, Teofana Teofana Puleva Puleva ** ** Danica Rosinová Rosinová ,, Peter * Institute of Automotive Mechatronics Slovak University of Technology in Bratislava, Slovakia (e-mail: * Institute of Automotive Mechatronics Slovak University of Technology in Bratislava, Slovakia (e-mail: * Slovak of [email protected]; [email protected]). * Institute Institute of of Automotive Automotive Mechatronics Mechatronics Slovak University University of Technology Technology in in Bratislava, Bratislava, Slovakia Slovakia (e-mail: (e-mail: [email protected]; [email protected]). [email protected]; [email protected]). ** Faculty of Automatics, Technical University of Sofia, Sofia, [email protected]; [email protected]). ** Faculty of Automatics, Technical University of Sofia, Sofia, ** of Technical University Bulgaria; (e-mail: [email protected]). ** Faculty FacultyBulgaria; of Automatics, Automatics, Technical University of of Sofia, Sofia, Sofia, Sofia, (e-mail: [email protected]). Bulgaria; Bulgaria; (e-mail: (e-mail: [email protected]). [email protected]). Abstract: The paper focuses on teaching advanced MIMO control topics using hydraulic system model. Abstract: The paper focuses on teaching advanced MIMO control topics using hydraulic system model. Abstract: focuses on teaching teaching MIMOare control topics using tank hydraulic Basic stepsThe of paper nonlinear MIMO system advanced control design shown on three plantsystem model.model. After Abstract: focuses on MIMO control topics using hydraulic Basic stepsThe of paper nonlinear MIMO system advanced control design are shown on three tank plantsystem model.model. After Basic steps of nonlinear MIMO system control design on three model. After modelling part, comprising nonlinear model based on are firstshown principles andtank its plant linearization about Basic steps of nonlinear MIMO system control design are shown on three tank plant model. After modelling part, comprising nonlinear model based on first principles and its linearization about modelling nonlinear model based on principles linearization about equilibriumpart, point,comprising two approaches to MIMO system control are demonstrated. Theitsfirst approach designs modelling nonlinear model based on first first principles and and linearization about equilibriumpart, point,comprising two approaches to MIMO system control are demonstrated. Theitsfirst approach designs equilibrium point, two approaches to MIMO system control are demonstrated. The first approach designs aequilibrium full state feedback PI controller in state space, the second one uses multi-loop approach with individual point, two approaches to MIMO system control are demonstrated. The first approach designs a full state feedback PI controller in state space, the second one uses multi-loop approach with individual aaloop full state PI in space, second one multi-loop approach individual design, overall stability condition is used tothe detune loop PIuses controllers for the overallwith system issues. fulldesign, state feedback feedback PI controller controller in state state space,to the second one multi-loop approach individual loop overall stability condition is used detune loop PIuses controllers for the overallwith system issues. loop design, overall stability condition is used to detune loop PI controllers for the overall system issues. The results are analysed and compared, important teaching issues are indicated. loop design,are overall stability condition isimportant used to detune loop PI controllers for the overall system issues. The results analysed and compared, teaching issues are indicated. The results are analysed and compared, important teaching issues are indicated. The results are analysed and compared, important teaching issues are indicated. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier rights reserved. Keywords: modelling, multi-input multi-output system, multi-loop control, Ltd. PI All controller, nonlinear Keywords: modelling, multi-input multi-output system, multi-loop control, PI controller, nonlinear Keywords: modelling, multi-input multi-output system, multi-loop control, PI controller, nonlinear system, robust stability Keywords: modelling, system, robust stability multi-input multi-output system, multi-loop control, PI controller, nonlinear system, system, robust robust stability stability 1. INTRODUCTION reliability improvement. Robustness is one of attractive 1. INTRODUCTION reliability improvement. Robustness is one of attractive 1. INTRODUCTION improvement. Robustness is one attractive qualities a multi-loop control scheme, such control Teaching advanced 1.control systems belongs to important reliability INTRODUCTION reliabilityof Robustness is since one of of attractive qualities ofimprovement. a multi-loop control scheme, since such control Teaching advanced control systems belongs to important structure qualities of a multi-loop control scheme, since such control can be inherently resistant to a wide range of Teaching advanced control systems belongs to important topics in mechatronics study. To connect theoretical concepts qualities of a multi-loop control scheme, since such control Teaching advanced control systems belongs to important structure can be inherently resistant to a wide range of topics in mechatronics study. To connect theoretical concepts structure can be inherently resistant to a wide range of uncertainties both in subsystems and interconnections. topics in mechatronics study. To connect theoretical concepts with applications, modelling and simulation of real plants structure can be inherently resistant to a wide range of topics in mechatronics study. To connect theoretical concepts uncertainties both in subsystems and interconnections. with applications, modelling and simulation of real plants uncertainties both in subsystems and interconnections. with applications, modelling and simulation of real plants contribute to increase interest and understanding of control uncertainties both in subsystems and interconnections. with applications, modelling and simulation of real plants contribute to increase interest and understanding of control We study the basic steps of nonlinear MIMO system control contribute to interest understanding control design fundaments, students to real of laboratory We study the basic steps of nonlinear MIMO system control contribute to increase increasepreparing interest and and understanding control design design fundaments, preparing students to real of laboratory We the of MIMO on three tanksteps process case study with 4system inputscontrol and 3 We study study the basic basic of nonlinear nonlinear MIMO design fundaments, preparing students to real laboratory work and practice, Oggunaike et al. (1994). Hydraulic design on three tanksteps process case study with 4system inputscontrol and 3 design and fundaments, real laboratory work practice, preparing Oggunaikestudents et al. to (1994). Hydraulic outputs. design on three tank process case study with 4 inputs and 3 After the modelling phase, control structure design on three tank process case study with 4 inputs and is work and practice, Oggunaike et al. (1994). Hydraulic systems belong to basic laboratory plants, used for simple as outputs. After the modelling phase, control structure is3 work and practice, Oggunaike al. (1994). systems belong to basic laboratoryetplants, used for Hydraulic simple as outputs. After the modelling phase, control structure is selected; a full size state feedback control is designed using outputs. aAfter the state modelling phase, control structure is systems belong basic plants, for as well as complex task demonstration, full size feedback control is designed using systems belong to to control basic laboratory laboratory plants, used usedRosinová for simple simpleand as selected; well as complex control task demonstration, Rosinová and selected; aa full size state feedback designed using pole placement technique in thecontrol state is space. Presented selected; full size state feedback control is designed using well as complex control task demonstration, Rosinová and Kozáková, (2009). Many control issues, including placement technique in the state space. Presented well as complex control task demonstration, Rosinová and pole Kozáková, (2009). Many control issues, including pole technique in space. Presented alternative – multi-loop requires a choice of pole placement placement technique control, in the the state state space. Presented Kozáková, (2009). Many control issues, nonlinearities and multi-input multi-output systemincluding control alternative – multi-loop control, requires a choice of Kozáková, (2009). Many control issues, including nonlinearities and multi-input multi-output system control appropriate alternative – multi-loop control, requires a choice input-output pairing. Further step is independent alternative input-output – multi-looppairing. control, requires a independent choice of of nonlinearities and multi-input multi-output system control can be taught and illustrated on various types of hydraulic appropriate Further step is nonlinearities multi-inputonmulti-output system control appropriate can be taught and and illustrated various types of hydraulic Further is single loopsinput-output design so pairing. that it guarantees of the appropriate input-output pairing. Further step stepstability is independent independent can be taught and illustrated on various types of hydraulic plants, as four tank system introduced by Johansson et al. single loops design so that it guarantees stability of the can be as taught on variousbytypes of hydraulic plants, four and tankillustrated system introduced Johansson et al. single so it of overall systemdesign including interactions. Two stability alternatives of single loops loops so that that it guarantees guarantees of the the plants, four system (1999), Johansson (2000). systemdesign including interactions. Two stability alternatives of plants, as as four tank tank system introduced introduced by by Johansson Johansson et et al. al. overall (1999), Johansson (2000). overall system including interactions. Two alternatives of stability condition for decentralized control structure are overall system including interactions. Two alternatives of (1999), Johansson (2000). stability condition for decentralized control structure are (1999), decentralized control structure are used: onecondition based on for small gain theorem for complementary In this Johansson paper, we(2000). consider a three tank system model stability stability condition for decentralized control structure are one based on small gain theorem for complementary In this paper, we consider a three tank system model used: one on small theorem for sensitivity function onegain for systems no RHP (right In wetheconsider aa three system corresponding laboratory plant tank delivered by model Inteco used: one based based onand small theorem with for complementary complementary In this this paper, paper, to three system sensitivity function and onegain for systems with no RHP (right corresponding towetheconsider laboratory plant tank delivered by model Inteco used: sensitivity function and one for systems with no RHP half plane) zeros. The obtained results are analysed and corresponding to plant delivered by company. Series of laboratory control design necessary for sensitivity and one for systems RHP (right (right corresponding to the the plant tasks delivered by Inteco Inteco plane)function zeros. The obtained resultswith are no analysed and company. Series of laboratory control design tasks necessary for half half plane) zeros. The obtained results are analysed compared. company. Series of control design tasks necessary for nonlinear MIMO controller design is presented. half plane) zeros. The obtained results are analysed and and company. Series system of control design tasks necessary The for compared. nonlinear MIMO system controller design is presented. The compared. nonlinear MIMO design is presented. The system system controlcontroller based on linearization around compared. nonlinear MIMO system controller design is presented. The 2. MODELLING AND CONTROL OF NONLINEAR nonlinear system control based on linearization around 2. MODELLING AND CONTROL OF NONLINEAR nonlinear system control based on linearization around working point is adopted. works only nonlinear system control Though based this on approach linearization around 2. AND CONTROL OF THREE SYSTEM working point is adopted. Though this approach works only 2. MODELLING MODELLING ANDTANK CONTROL OF NONLINEAR NONLINEAR THREE TANK SYSTEM working point is adopted. Though this approach works only within limited operation region, its simplicity and working point is adopted. Though this approach works only THREE TANK SYSTEM within limited operation region, its simplicity and In this section, key pointsTANK of teaching basics of nonlinear THREE SYSTEM within operation region, its simplicity applicability in many real plants attractive and In this section, key points of teaching basics of nonlinear within limited limited operation region,make its it applicability in many real plants make itsimplicity attractive and and In points of nonlinear modeling and key control, three basics tank of model, are In this this section, section, pointsusing of teaching teaching nonlinear applicability in many real plants make it attractive and frequently used in nonlinear systems control. Modelling part and key control, using three basics tank of model, are applicability in inmany real systems plants make it Modelling attractive part and modeling frequently used nonlinear control. modeling and using tank model, are summarized. The control, aim of this part ofthree advanced control course modeling and control, using three tank model, are frequently used in nonlinear systems control. Modelling part comprises nonlinear model and its linearization around summarized. The aim of this part of advanced control course frequently used in nonlinear Modelling part summarized. comprises nonlinear model systems and itscontrol. linearization around The aim of this part of advanced control course is to cover basic steps necessary to design appropriate control summarized. The aim of this part of advanced control course comprises model and itsslightly linearization equilibrium point. (The results differentaround from is to cover basic steps necessary to design appropriate control comprises nonlinear nonlinear model andare linearization equilibrium point. (The results areitsslightly differentaround from is basic necessary design forto acover nonlinear MIMO system,to realized and verifiedcontrol on a basic steps steps necessary design appropriate appropriate equilibrium point. (The results are slightly different from those presented in plant user manual.) The main focus is on is fortoacover nonlinear MIMO system,torealized and verifiedcontrol on a equilibrium point. (The results are slightly different from those presented in plant user manual.) The main focus is on for a nonlinear MIMO system, realized and verified simulation model of a three tank system. for a nonlinear system, realized and verified on on aa those presented plant user manual.) The main focus is on controller designin and comparison of different control design simulation modelMIMO of a three tank system. those presented in plant user manual.) The main focus is on controller design and comparison of different control design simulation model of a three tank system. model of a three tank system. controller design comparison of control approaches. Stateand feedback control with full sizedesign gain simulation part includes: controller design comparison of different different control approaches. Stateandfeedback control with full sizedesign gain Modelling Modelling part includes: approaches. State feedback control with full size gain matrices is confronted with a multi-loop controller (called part includes: approaches. State feedback withcontroller full size(called gain Modelling matrices is confronted with a control multi-loop Modelling of part includes:three tank mathematical model, input nonlinear matrices is with multi-loop controller (called also decentralized controller, some sources). matrices is confronted confronted with aa according multi-loopto (called -- analysis analysis of nonlinear three tank mathematical model, input also decentralized controller, according tocontroller some sources). -- analysis nonlinear mathematical outputof variables andthree theirtank ranges, also according some Multi-loop control controller, reduces multivariable problem and analysis nonlinear mathematical model, model, input input also decentralized decentralized according to tocontrol some sources). sources). outputof variables andthree theirtank ranges, Multi-loop control controller, reduces multivariable control problem and and output variables and their ranges, Multi-loop control reduces multivariable control problem complexity, and enables to employ decentralized control and output variables and their ranges, Multi-loop control reducesto multivariable control problem complexity, and enables employ decentralized control - determination of working area – steady state of nonlinear complexity, enables employ structure withand subsystems having theirdecentralized local control control loops. - determination of working area – steady state of nonlinear complexity, enables to to employ structure withand subsystems having theirdecentralized local control control loops. --system determination of working and control determination of aim working area area – – steady steady state state of of nonlinear nonlinear structure with local control loops. Compared withsubsystems centralized having MIMO their controller systems, multi- system and control aim structure with subsystems having their local control Compared with centralized MIMO controller systems, loops. multi- system and control aim system and control aim Compared with centralized MIMO controller systems, multiloop control structure generally yields certain performance Compared with centralized MIMOyields controller systems, multi- - derivation of linearized model using standard Taylor series loop control structure generally certain performance - derivation of linearized model using standard Taylor series loop control structure yields certain deterioration, however,generally weighted against byperformance important approach loop control structure generally yields certain performance -- derivation deterioration, however, weighted against by important approach derivation of of linearized linearized model model using using standard standard Taylor Taylor series series deterioration, however, weighted against by important benefits, such as design simplicity, hardware, operation and deterioration, however, weighted against by important approach benefits, such as design simplicity, hardware, operation and approach benefits, benefits, such such as as design design simplicity, simplicity, hardware, hardware, operation operation and and
2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 IFAC 360Hosting by Elsevier Ltd. All rights reserved. Copyright 2016 responsibility IFAC 360Control. Peer review©under of International Federation of Automatic Copyright © 2016 IFAC 360 10.1016/j.ifacol.2016.07.204 Copyright © 2016 IFAC 360
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- verification of linearized model: comparison of nonlinear and linear model using the respective Simulink schemes; importance of settling the system into the working point
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difficulties in achieving the control aim - a high accuracy control of the tank levels.
Controller design part consists of:
6 1
- determination of control structure (full size controller versus multi-loop – decentralized controller)
7
- choosing appropriate input-output pairing for multi-loop controller
2
- loop controller design (PI controllers) 3
- testing (robust) stability condition under decentralized controller design - verifying the designed controllers by simulation in Simulink, comparison of linearized model and nonlinear model results.
5
4
Fig. 1 Physical model of multitank system 3.1 Nonlinear model of three tank system
Presented material can be preferrably used for project oriented teaching, when the students have to complete tasks and write a report, summarizing all procedures and results. The important part of the project is analysis of the obtained results and comparison of the designed controllers and closed loop responses both for original nonlinear and linearized model.
The mathematical model of multi tank system can be derived from the general model of n-tanks. The model of the process dynamics can be obtained by the mass balance equation dV1 dH1 q C1H1 1 wl1H1 1 dH1 dt
dV2 dH 2 C1H1 1 C2 H 2 2 wl1H1 1 wl 2 H 2 2 dH 2 dt
3. THREE TANK SYSTEM MODELLING The model of multi tank system, corresponding to the one, produced by Inteco Co is considered, Fig.1. It comprises three separated tanks placed above each other and interconnected with drain valves (Inteco Co., 2006). The upper tank has a constant cross section, while the others have a variable cross section: trapezoidal middle one and round lower one. These variable cross sections add further nonlinearities in the system, besides physical laws of flow. The water is pumped into the upper tank from the supply tank by a pump driven by a DC motor. The water outflows from the tanks due to gravity through orifices placed in the bottom of tanks. The tank valves act as flow resistors. The area ratio of the valves is controlled and can be used to vary the outflow characteristic. Each tank is equipped with a level sensor based on hydraulic pressure measurement.
(1)
dVn dH n Cn 1H n n11 Cn H n n wl ( n 1) H n n11 wln H n n dH n dt where Vi is the fluid volume of the i-th tank, Ci is the resistance of the i-th output orifice, wli is the resistance of the
i-th disturbance orifice and H i is the level in the i-th tank. For the laminar flows, the outflow rate from a tank can be described by the Bernoulli law with i 1/ 2. For real
configuration, a more general coefficients i are applied. From (1), the nonlinear multi tank system model is obtained dH1 1 1 1 q CH 1 w H 1, (2) dt 1 ( H1 ) 1 ( H1 ) 1 1 1 ( H1 ) l1 1
The standard procedure of the model development is adopted, consisting of the following steps: developing the mathematical model based on the basic physical laws; determination of equilibrium respective to the working point and the respective model linearization; building the simulation model and its verification.
dH 2 1 1 1 C1H1 1 C2 H 2 2 w H 1 , (3) dt 2 ( H 2 ) 2 ( H 2 ) 2 ( H 2 ) l1 1
dH 3 1 1 C H 2 C H 3 3 ( H 3 ) 2 2 3 ( H 3 ) 3 3 dt 1 1 wl 2 H 2 2 w H 3 3 ( H 3 ) 3 ( H 3 ) l 3 3
The nonlinear model is developed in the state space, where liquid levels H 1 , H 2 , H 3 in the tanks are measurable state variables of the system. The controlled inputs are: liquid flow rate q and valves settings C1 , C 2 , C3 . Output variables are the same as state ones, which later simplifies state space controller design. In Fig. 1, the multi tank system components interconnection is shown (Inteco, 2006). In this paper the control strategy based on pump and valves control system is investigated. The nonlinearities caused by shapes of tanks and those introduced by valve flow dynamics cause
1 ( H1 ) aw
2 ( H 2 ) cw
H2
H 2 max
bw
,
(4)
(5)
2 2 3 ( H 3 ) w R (R H3 )
where i ( H i ) is the cross sectional area of i-th tank at the level H i . 361
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As mentioned above, the middle and lower tanks have a variable cross sectional area. We assume for simplicity zero disturbances wl1 wl 2 wl 3 .
respective equilibrium, therefore the initial conditions of used integrators must be appropriately preset. The results of nonlinear an linear model for H1 and H2 are shown in Fig.2, result for H3 is similar (step change from equilibrium was performed in time 500).
3.2 Linearized model of the three tank system The linearized models of the upper, middle and lower tanks are obtained by the Taylor expansion of (2)-(4) around the assumed equilibrium state represented by the defined pump flow rate, valves resistance and the tank water levels denoted as q0 , Ci 0 , i 1,3 and Hi 0 , i 1,3 respectively. The state
Table 1. Geometrical parameters of the tanks Upper tank
vector, the control signal, and the output are:
x H1 H 2 H 3 T , u q C1 C2 C3 T ; y=x. (6) In the equilibrium point, the input flow rate of the i-th tank equals to its output flow rate
10 20 q0 C C C30 H3030 . 10 H10 20 H 20
(7)
The state space description of the system linearized around the defined equilibrium point is
x Ax Bu
, y Cx where the components of the state matrix A are given by a Jacobian matrix respective to (2)-(4): C H 1 1 a11 10 1 10 , a12 a13 0 , 1 ( H10 )
H1max 0.35 a = 0.25 m w= 0.035 m
Pump 1.4.104
(8)
Table 3. Level references
linearized model nonlinear model 0.135
0.13
0.125
32 ( H 30 )
0.12 400
.
800
1000
t
1200
1400
1600
0.12
0.115
3
H 30 H 20 , b34 , 3 ( H 30 ) 3 ( H 30 )
H2
linearized model nonlinear model 0.11
1 2 H10 H 20 , b23 , 2 ( H 20 ) 2 ( H 20 ) 2
600
0.125
1 H10 1 , b12 , b b 0, 1 ( H10 ) 13 14 1 ( H10 )
b31 b32 0 , b33
Lower tank 0.1 m
0.14
22 ( H 20 )
3 3 1 2 ' 20 H 20 C30 H 30 3 ( H 30 ) C30 3 H 30 3 ( H 30 )
b21 b24 0 , b22
Middle tank 0.1 m
0.15
The elements of the matrix B(3 x 4) are
b11
Lower 5.5.105
0.12 m
C H 2 1 a31 0 , a32 20 2 20 , 3 ( H 30 )
a33
Middle 5.5.105
0.145
Upper 5.5.105
Upper tank
1 2 2 1 ' 10H10 C20 H 20 2 ( H 20 ) C20 2 H 20 2 ( H 20 )
C
H3max 0.35 m R = 0.364 m w= 0.035 m
3 Valves [m / s]
[ m3 / s ]
H1
a22
Lower tank
Table 2. Flow rate limits
C H 1 1 a21 10 1 10 , a23 0 , 2 ( H 20 )
C
Middle tank H 2max 0.35 m b = 0.345 m c = 0.1 w= 0.035 m
0.105
0.1 400
(9)
and C is identity matrix I (3 x3) .
600
800
1000
t
1200
1400
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Fig.2 Model responses for linearized and nonlinear model 4. CONTROLLER DESIGN The main aim of this part is to confront two basic possibilities for MIMO system controller design based on material taught in control courses. In both cases, PI controllers are designed to remove steady state control error.
In Tables 1 and 2, the geometrical parameters and technological limits of the control signals are given. The level references are shown in Table 3, the corresponding steady state pump flow rate is q0 2,4.105 m3 / s .
a) Full MIMO controller designed in state space by pole placement method (Slavov and Puleva (2012))
The linearized model is verified and compared with the nonlinear one in Simulink environment. It is important to underline that verification must be performed around the
b) Multiloop (decentralized) controller. 362
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4.1 Full state feedback controller
0.16 0.14
The first option uses Matlab function place for an augmented system (10), including the additional integral states.
0.12
linear system nonlinear system
(10)
H1
0.1
B A 0 33 x x u 0 34 C 0 33 y C 0 33 x
363
0.08 0.06
0.04
where x xT xiT denotes the augmented state.
0.02
The resulting controller can be divided into its proportional part (feedback gain) and integral part placed in the forward path, see Fig.3.
0
0
500
t
1000
1500
0.14
1 s
0.12
Integrator1
Integrator2 1 s Integrator3
K*u Ki
0.1
x' = Ax+Bu y = Cx+Du
output
State-Space1
To Workspace4
linear system nonlinear system
H2
1 s
K*u
0.08
0.06
Kp 0.04
0.02
Fig. 3. Closed loop with full state feedback PI controller To ensure tank system step response without overshoot and a settling time (up to 200s), the desired poles are chosen as p -0.0333 -0.0367 -0.0400 -0.0433 -0.0467 -0.0500 The obtained controller matrices are 0.1602 0.0795 0.0359 -0.0378 0.1655 0.0650 4 Ki 10 , -0.0222 -0.0920 0.0733 -0.0225 -0.0928 -0.1470 0.7058 0.3604 0.1803 -0.0652 0.7006 0.3264 3 10 . K -0.1424 -0.3184 0.3681 -0.1441 -0.4748 -0.7386 Simulation results for both linear and nonlinear systems with the designed full MIMO PI controller are shown in Fig. 4. Several observations can be made by studying the responses of linear and nonlinear model. These observations belong to crucial student tasks in project completing. While the responses of linearized model does not depend on the working point, the nonlinear model have different behaviour in different operating ranges. The first part of responses up to 500s belong to transient into equilibrium point respective to linearization. After reaching the operating point corresponding to the reference levels in all tanks, listed in Table 3, step responses of the designed control loops are studied: 10% step changes of reference levels are made in 500s for H1, in 700s for H2, and, finally, in 900s for H3. It should be noted that in the working range (after the reference levels were reached), the linearized system dynamics is closer to the nonlinear one, than within transient time up to 500s.
0
500
t
1000
1500
0.13 0.12 0.11 0.1 linear system nonlinear system
H3
0.09 0.08 0.07 0.06 0.05 0.04 0.03
0
500
t
1000
1500
Fig. 4. Comparison of closed loop step responses for a linearized and nonlinear three tank model with full state feedback PI controller 4.2 Multi-loop controller The second controller design approach addresses the multiloop controller and is carried out in frequency domain. The transfer function matrix G(s), corresponding to the linearized model (8), (9) is obtained by standard Matlab conversion. To start a multi-loop controller design, it is inevitable to appropriately choose input-output pairing. In the above three tank model, 4 inputs and 3 outputs are considered. The first task is then to choose 3 of the existing inputs appropriate for a multi-loop design. This can be done analysing the steady state gain matrix. The appropriate choice is pairing input flow q0 to output H1, input valve resistance C20 to output H2, and input valve resistance C30 to output H3, thus omitting the 363
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second input C10. Then, the multiloop controller design is performed for a square transfer function matrix G(s). 114.3 0 0 s 0.0114 45.5 1.644 G( s ) 2 0 s 0.0287 s 0.0002 s 0.0173 36.05 0.0225 36.05s 3 s 0.0424s 2 0.0006s s 2 0.03 s 0.0002 s 0.0137
low frequencies, where TD 1 , for stable system with no
(10)
(11)
E ( G GD )G 1 GM G 1 , SD ( I GD R )1 . The above condition can be reformulated as:
G 1 S D M 0
To find R( s ) , the so called independent design is considered, where individual loops are designed “independently” (simultaneously). Local controllers Ri ( s ) are designed so that they:
(15)
The multiloop controller was designed independently for diagonal transfer functions from (10), resulting controllers were obtained after detuning to satisfy stability condition (15), see Fig. 5. Stability condition (14) is not appropriate since its module is significantly below 1 for low frequencies. Loop controllers for (11) are
b) satisfy the overall system stability condition. In the following, sensitivity is denoted as 1 S( s ) ( I G( s )C( s )) and closed loop transfer function (complementary sensitivity) is denoted as T ( s ) G( s )C( s )( I G( s )C( s )) 1 .
R1( s )
Let us turn to stability condition for a system with multiloop controller. Matrix G(s) can be splitted into its diagonal and off-diagonal parts: G( s ) GD ( s ) GM ( s ) .
8.75s 0.1 4 44s 0.75 4 .10 , R2 ( s ) .10 , s s
R3 ( s )
41.6s 0.57 .10 4 s
1.4
The potential uncertainties (due to linearization) can be included into GM ( s ) . For a stable open loop system G(s)R(s), the closed loop system stability condition based on small gain theorem is given in Veselý and Harsányi (2008).
1.2 1 0.8
Lemma 1
0.6
Consider stable system G(s) with a decentralized controller C(s). The respective closed loop system T(s) is stable if
0.4
abs(S1) abs(S2) abs(S3) 1/norm(Gm*G)
0.2
GD1W GM 1
(12)
or
1 GM
(13)
where matrix W is given by R 1 GD GDW 1 . Inequality (13) can be reformulated into
GD1TD M 0
1 . GM
Either of alternatives (14) or (15) must be satisfied for all frequencies.
a) stabilize individual loops
GD1W
Lemma 2 (Skogestad and Postlethwaite, 2009) Consider stable system G(s) with decentralized controller R(s). Assuming that neither G nor GD has RHP zeros, the 1 overall closed loop system is stable if and only if I ES D is stable, where
A multi-loop PI controller is considered: 1 0 0 P1 T1s 0 0 R1( s ) 1 R( s ) 0 R2 ( s ) P2 0 0 0 T2 s R3 ( s ) 0 0 1 0 P3 0 T3 s
RHP zeros this may be too restrictive. The alternative condition for this case is in Lemma 2.
1 GM
(14)
where TD GD R( I GD R )1 . Condition (14) can be used for stable system without or with RHP zeros (both for minimum and non-minimum phase case). However, the above condition can be rather limiting in 364
0 -4 10
(6a) -3
10
-2
10 frequency
-1
10
0
10
Fig. 5. Overall system stability condition for a multiloop controller. Red line denotes the upper limit to keep stability. Significant point of controller design task is a comparison of responses obtained for full MIMO controller from Section 4.1and responses of multi-loop controller. Step responses in Fig. 6 indicate that multiloop controller outperforms the former one, however, is more sensitive. Control inputs range is similar, though, in multiloop case, rapid changes of control variables appear, Fig. 7. Detail performance indices summary is omitted due to space reasons.
2016 IFAC ACE June 1-3, 2016. Bratislava, Slovakia
Danica Rosinová et al. / IFAC-PapersOnLine 49-6 (2016) 360–365
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-5
x 10
0.16
6
0.14 0.12
4 full MIMO controller multiloop controller
2
0.08
u
H1
0.1
0
0.06
-2
0.04
-4
0.02 0
0
500
t
1000
-6
1500
0
100
200
300
t
400
500
600
700
0.13
(9)
0.12 -5
0.11
1
x 10
0.1
0
0.09
H2
full MIMO controller multiloop controller
0.08
-1
0.07
-2 u
0.06
-3
0.05 0.04
0
500
t
1000
-4
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-5 -6
0.13 0.12
0
500
t
1000
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0.11 0.1
H3
0.09
Fig. 7 Control input responses for a full MIMO controller (figure above) and multiloop controller (figure below)
full MIMO controller multiloop controller
0.08
The work has been supported by Grant No 030STU-4/2015 of the Cultural and Educational Grant Agency of the Ministry of Education, Science, Research and Sport of the Slovak Republic.
0.07 0.06 0.05 0.04 0.03
0
500
t
1000
REFERENCES Johansson, K. H., Horch, A., Wijk, O. and Hansson, A. (1999). Teaching Multivariable Control Using the Quadruple-Tank Process. In: Proc. 38nd IEEE CDC, Phoenix, AZ Johansson, K.H. (2000). The Quadruple-Tank Process: A Multivariable Laboratory Process with an Adjustable Zero. IEEE Transactions on Control Systems Technology, Vol. 8, No. 3, pp.456-465. Multitank system, User’s Manual, Inteco Co, 2006. Ogunnaike, Babatunde A and Harmon W Ray (1994). Process dynamics, modeling, and control. New York: Oxford University Press. ISBN 01-950-9119-1. Rosinová, D. and Kozáková, A. (2009). Robust decentralized PID controller design: a case study. In: IEEE Sankt Peterburg, Russia Skogestad, S. and Postlethwaite, I. (2009). Multivariable feedback control: analysis and design. John Wiley & Sons Ltd., Chichester, West Sussex, UK. Veselý, V. and Haršányi, L. (2008). Robust control of dynamic systems. Vydavateľstvo STU, Bratislava, SR.
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Fig. 6. Step responses of output tank levels for a full MIMO controller and multiloop controller 5. CONCLUSION Control design for a three tank multi-input multi-output system is analysed in this paper from a teaching point of view. The adopted control design approach is based on linearized model of originally nonlinear system around the working point. Steps of the overall control design procedure are indicated including modelling, the corresponding student tasks are proposed. Two approaches: centralized and (8) decentralized control, are compared. Skills gained using control design based on linearized model are very useful also for further studies of LPV (Linear Parameter Varying), gain scheduling and other, more general control schemes for nonlinear systems.
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