- Email: [email protected]
A Multivariable QFT Design for a Paper Machine Benchmark
7c-024
Copyright © 1996 IFAC 13th Triennial World Congress. San Francisco. USA
A MULTIVARIABLE QFT DESIGN FOR A PAPER MACHINE BENCHMARK Mattias Nordin
Optimization and Systems Theory Royal Institute of Technology 100 44 Stockholm, Sweden [email protected]
Abstract. This paper presents an alternative robust control system design for an uncertain non-linear multivariable paper machine benchmark problem. The aim is to control the output variables basis weight and ash contents with the control inputs thickstock and filler flow, such that time domain specifications are fullfilled for all plant cases. In this paper, a static non-diagonal non-linear precompensator is first designed to minimize steady state cross coupling. Then a set of equivalent linear transfer functions are identified by Least Squares. The time domain specifications are translated to approximative frequency domain specifications .. Based on the frequency domain specifications, and the set of equivalent transfer functions, a diagonal dynamic feedback controller and a diagonal dynamic prefilter are designed according to the Horowitz robust. control design method, or QFT. Time domain simulations of the controlled non-linear system show that the specifications are satisfied. The paper illustrates a practical design method for multivariable paper machine control, whereby design trade-offs are made visible during the design process. Keywords. Multivariable Control, Robust control, Robust Performance, Benchmark Examples
1. INTRODUCTION
At the conference Control Systems 94, Stockholm, Sweden (Edlund, 1994) a paper machine benchmark was presented, together with several control system design suggestions (Hagberg and Isaksson, 1994; Wailer, 1994; Makkonen et al., 1994; Piipponen and Ritala, 1994; Ebach and Griiser, 1994; Whidborne et al., 1994; Bozin and Austin, 1994; Chow et al., 1994; Fu and Dumont, 1994). The benchmark and a number of designs has also been published in Control Engineering Practice, October 1995. These designs were essentially of three types: Smith predictor control, predictive control - dynamic matrix control and generalized predietive control and a
robust Hoo loop shaping. None of them did however explicitly compensate for the plant uncertainty, which essentially was in one parameter only, namely the retention. In some designs there were some attempts to adapt for this parameter, or its effect. The only robust controller design, uses unstructured uncertainty on normalized coprime factor form. In this paper a robust linear controller for a set of linear uncertain plants is designed. These plants are identified from simulations of the non-linear paper machine model, and not by including e.g. unstructured uncertainty to a nominal case, whieh tends to yield conservative control designs. The design satisfies the time domain specifica-
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tions, with a controller of low degree. The design procedure, and the paper is organized as follows: In section 2 the plant is 'linearized' and 'diagonalized' with a nonlinear, static, transformation of the input vector. Thereafter the dynamics of the 2 by 2 paper machine is analyzed via Least Squares identification, whereby the retention assumes various parameter values. The result is a set of linear transfer functions. For each frequency the set of linear transfer functions defines a complex value set, or template. Some templates are plotted. In section 3 the time domain specifications are translated to approximative frequency domain specifications, according to the procedures in (Horowitz, 1992). In section 4 the controllers are designed and simulated. To achieve robust performance for all plant cases, Horowitz bounds, or constraints, for the controller or the nominal open loop are calculated.With these the design is performed by manual loop shaping in the Nichols and Bode charts, loop by loop, and prefilters are designed for the reference step specifications. The design is simulated, both for the identified linear models, and for the full 22 state nonlinear paper machine model. In section 5 the results are summarized, the advantages of the proposed design are discusses. It is also speculated how the robust design could be complemented with adaptation.
2. THE PAPER MACHINE MODEL SPECIFICATIONS AND IDENTIFICATION The paper machine model is taken from a benchmark that was first presented at the Conference on Control Systems in the Pulp and Paper Industry, May 94, Stockholm, Sweden, (Edlund, 1994). A model of a paper machine with 22 states is introduced. The outputs are measurements of dry weight and ash content, sampled every 30 seconds. The control inputs are filler valve level VI and thickstock valve level V2. The dynamics of the model is not analytically analyzed in this paper. Rather the plant is considered as a black box model, depending on one parameter, the retention level R. The objective is to control dry weight and ash content 'as well as possible', given the two inputs filler and thickstock flow. In the benchmark problem 'as well as possible' is given as a set of specified set point and disturbance responses, that should be fulfilled for all retention levels in the range [0.3, ll.8]. We have chosen to present a subset of the testcases, although the QFT design method is suitable for all of them. The following
time domain specifications, corresponding to test cases 1,2 and 4 in the benchmark, were used: (1) A reference step, 55 g/m 2 to 60 gjm 2 , in dry weight. The overshoot should be less than 10% and the design objective is to minimize the settling time Ts (5 %). The ash content should not deviate more than 0.3 units from the setpoint of 20 %. (2) A reference step, 15% to 20%, in ash content. the overshoot should be less than 10% and the design objective is to minimize t.he settling time Ts (5 %). The dry weight should not deviate more than 1 units from the setpoint 60 gjm 2 . (3) A step disturbance in consistency before the headbox. No specification other than to minimize its effect. First the nonlinearity of the system is changed by transforming the inputs filler flow VI and thickstock flow V2 to mass flow and ash content, in the following non-linear way: VI
= (Ul + 0.17u2)(0.07026u2 -
V2
0.626)
= (111 + 0.17112)(1.21- 0.0121112)
(1)
The new inputs U1 and U2 now corresponds to dry weight and ash content respectively, and the system is now linear in steady state The constants in the transformation v = T (11) were chosen from open loop steady state simulations, such that the non-linearly transformed system is as diagonal as possible, for all plant cases. A block diagram of the plant with the non-linear input transformation, and the constant matrix precompensation is found in figure 1. .-_ _ _----;filler valve
_u_1__ :1
~
T(u)
I::
dry weight
:I,---Pl_an_t--.-JI
thickstock valve
: :: ~w~m
Fig. 1. Block diagram of the transformed plant. The main uncertainty of the system is the variation of the retention level in the interval [0.3,0.8]. To get an accurate description of the effect of this uncertainty, the transformed system is simulated in open loop near the operating point, with PRBSs having the approximate amplitude of the benchmark step tests. This was performed for 25 equidistant retention levels. The inputoutput pairs were used to identify 25 linear 2 by 2 models, y(z) = pR(z)u(z) where :tIl is dry weight and Y2 is ash content, with a standard RLS identification scheme, (Ljung, 1991). In figures 2 and 3 Bode plots of the 25 identified models are shown, together with templates for some frequencies. Since the 25 identified model were
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~/O
The MIMO block diagram can be transformed into 4 equi valent SIM 0 loops. We first design 91 (z) i.e. the first loop. Since the system is a multivariable system, also 92 (z) has effect on this loop. It is however possible to estimate the effect of 92 with respect to the first loop with the frequency domain bounds for loop 2. Note that this is not equivalent to assuming that 92 = 0, and closing the system loop by loop. To achieve robust performance for the first loop we calculate Horowitz bounds that a nominal open loop must fulfill, and then design 91 (z) by manual loop shaping. The transfer function 91 (z)
92(Z) = 0.83z - 0.47 0.953z - 0.88 z- 1 z - 0.84 1.54z 2 - 1.375z + 0.673 0.642z 2 + 0.619z - 0.0227(4) Z2 - 0.227z + 0.064 Z2 - 0.124z + 0.114
In the lower half of figure 6 the bounds and the nominal open loop is shown. Notice again that the bounds are not completely satisfied. Given 92(Z) the prefilter Jz(z) is designed, so that the frequency response specifications are fulfilled, see figure 4: Jz(z) = 1.31z - 1.2 0.374z2 - 0.245z + 0.118
=
0.54z - 0.36 1.23z + 0.583 z - 1 z2 - 0.785z + 0.236 0.536z 2 + 0.517 z - 0.019 Z2 - 0.7852z + 0.236
1.lz 2 -
z - 0.886 (2)
was finally chosen. In the upper half of figure 6 the bounds and the nominal open loop is shown. Bounds and nominal system for loop 1
40
2°5~~;;;;;:::~~"".~15JJ~ . ~::;3~~~::::~~ -20r-======----::========""'............. o -350 -300 -250 -200 -150 -100 -50
Z2 -
Z
(5)
+ 0.253
It is not a serious shortcoming that the bounds are not completely satisfied, since the bounds emanate from the approximate frequency domain specification. The bounds give however an indication of what frequencies might contribute to difficulties in satisfying the timedomain specifications. Bode plots of the controllers are shown in figure 7.
•~t
M8!J1i1ude of 91 and
92 (dotted)
~--.--.-.-o-r---~-----.--.--.--........ . .. 10--3
10-2
[rad/sl
Phase of 91 and g2 (dottid)
Bounds and nominal system for loop 2
i
_~:i:=:::::::1~,=====:~--.--.-.-o-r---.",-,-,-::---... .
-------.---.--..---....:::1.... ..
10-2 [radls) Ma",itude of 11 and 12 (cloned)
10-3
10- 1
(radls]
Fig. 6. Horowitz bounds for specifications 1 and 2
Fig. 7. Bode plots of 91 and 92 and magnitude plots of JI and Jz
Notice that it is impossible to satisfy the Horowitz bound constraint, due to the delay in the plant that limits the bandwidth. Given 9dz) the prefilter JI(z) is designed, so that the frequency response specifications are approximately fulfilled, see figure 4: JI(z) = 2.9z - 2.39 0.412z2 - 0.368z + 0.18 z - 0.49 z2 - 0.848z + 0.0723
(3)
Given 91 (z) it can be included in the plant, and the remaining design is a SIMO problem. Horowitz bounds that the second loop must fulfill for a nominal case can now be calculated. Loop shaping of 92(Z) gives
The design must be checked in the time domain. Simulations for the linear equivalent systems in closed loop are given in figure 8. Notice that all specifications are satisfied. The achieved settling time for all retention cases are TB < 310 for specification 1 and Ts < 330 for specification 2. This is slower than some of the proposed designs in (Edlund, 1994), but none of them fulfills the overshoot specifications for all ret.entions! For specification 3 it can be noted that there is no oscillatory behavior for any of the retention levels and that the disturbance rejection is as good as any of the previously suggested controllers. Simulations for the non-linear plant controlled with G(s) and F(s), according to block diagram figure 5,
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Copyright © 1996 IFAC 13th Triennial World Congress. San Francisco. USA
A MULTIVARIABLE QFT DESIGN FOR A PAPER MACHINE BENCHMARK Mattias Nordin
Optimization and Systems Theory Royal Institute of Technology 100 44 Stockholm, Sweden [email protected]
Abstract. This paper presents an alternative robust control system design for an uncertain non-linear multivariable paper machine benchmark problem. The aim is to control the output variables basis weight and ash contents with the control inputs thickstock and filler flow, such that time domain specifications are fullfilled for all plant cases. In this paper, a static non-diagonal non-linear precompensator is first designed to minimize steady state cross coupling. Then a set of equivalent linear transfer functions are identified by Least Squares. The time domain specifications are translated to approximative frequency domain specifications .. Based on the frequency domain specifications, and the set of equivalent transfer functions, a diagonal dynamic feedback controller and a diagonal dynamic prefilter are designed according to the Horowitz robust. control design method, or QFT. Time domain simulations of the controlled non-linear system show that the specifications are satisfied. The paper illustrates a practical design method for multivariable paper machine control, whereby design trade-offs are made visible during the design process. Keywords. Multivariable Control, Robust control, Robust Performance, Benchmark Examples
1. INTRODUCTION
At the conference Control Systems 94, Stockholm, Sweden (Edlund, 1994) a paper machine benchmark was presented, together with several control system design suggestions (Hagberg and Isaksson, 1994; Wailer, 1994; Makkonen et al., 1994; Piipponen and Ritala, 1994; Ebach and Griiser, 1994; Whidborne et al., 1994; Bozin and Austin, 1994; Chow et al., 1994; Fu and Dumont, 1994). The benchmark and a number of designs has also been published in Control Engineering Practice, October 1995. These designs were essentially of three types: Smith predictor control, predictive control - dynamic matrix control and generalized predietive control and a
robust Hoo loop shaping. None of them did however explicitly compensate for the plant uncertainty, which essentially was in one parameter only, namely the retention. In some designs there were some attempts to adapt for this parameter, or its effect. The only robust controller design, uses unstructured uncertainty on normalized coprime factor form. In this paper a robust linear controller for a set of linear uncertain plants is designed. These plants are identified from simulations of the non-linear paper machine model, and not by including e.g. unstructured uncertainty to a nominal case, whieh tends to yield conservative control designs. The design satisfies the time domain specifica-
6692
tions, with a controller of low degree. The design procedure, and the paper is organized as follows: In section 2 the plant is 'linearized' and 'diagonalized' with a nonlinear, static, transformation of the input vector. Thereafter the dynamics of the 2 by 2 paper machine is analyzed via Least Squares identification, whereby the retention assumes various parameter values. The result is a set of linear transfer functions. For each frequency the set of linear transfer functions defines a complex value set, or template. Some templates are plotted. In section 3 the time domain specifications are translated to approximative frequency domain specifications, according to the procedures in (Horowitz, 1992). In section 4 the controllers are designed and simulated. To achieve robust performance for all plant cases, Horowitz bounds, or constraints, for the controller or the nominal open loop are calculated.With these the design is performed by manual loop shaping in the Nichols and Bode charts, loop by loop, and prefilters are designed for the reference step specifications. The design is simulated, both for the identified linear models, and for the full 22 state nonlinear paper machine model. In section 5 the results are summarized, the advantages of the proposed design are discusses. It is also speculated how the robust design could be complemented with adaptation.
2. THE PAPER MACHINE MODEL SPECIFICATIONS AND IDENTIFICATION The paper machine model is taken from a benchmark that was first presented at the Conference on Control Systems in the Pulp and Paper Industry, May 94, Stockholm, Sweden, (Edlund, 1994). A model of a paper machine with 22 states is introduced. The outputs are measurements of dry weight and ash content, sampled every 30 seconds. The control inputs are filler valve level VI and thickstock valve level V2. The dynamics of the model is not analytically analyzed in this paper. Rather the plant is considered as a black box model, depending on one parameter, the retention level R. The objective is to control dry weight and ash content 'as well as possible', given the two inputs filler and thickstock flow. In the benchmark problem 'as well as possible' is given as a set of specified set point and disturbance responses, that should be fulfilled for all retention levels in the range [0.3, ll.8]. We have chosen to present a subset of the testcases, although the QFT design method is suitable for all of them. The following
time domain specifications, corresponding to test cases 1,2 and 4 in the benchmark, were used: (1) A reference step, 55 g/m 2 to 60 gjm 2 , in dry weight. The overshoot should be less than 10% and the design objective is to minimize the settling time Ts (5 %). The ash content should not deviate more than 0.3 units from the setpoint of 20 %. (2) A reference step, 15% to 20%, in ash content. the overshoot should be less than 10% and the design objective is to minimize t.he settling time Ts (5 %). The dry weight should not deviate more than 1 units from the setpoint 60 gjm 2 . (3) A step disturbance in consistency before the headbox. No specification other than to minimize its effect. First the nonlinearity of the system is changed by transforming the inputs filler flow VI and thickstock flow V2 to mass flow and ash content, in the following non-linear way: VI
= (Ul + 0.17u2)(0.07026u2 -
V2
0.626)
= (111 + 0.17112)(1.21- 0.0121112)
(1)
The new inputs U1 and U2 now corresponds to dry weight and ash content respectively, and the system is now linear in steady state The constants in the transformation v = T (11) were chosen from open loop steady state simulations, such that the non-linearly transformed system is as diagonal as possible, for all plant cases. A block diagram of the plant with the non-linear input transformation, and the constant matrix precompensation is found in figure 1. .-_ _ _----;filler valve
_u_1__ :1
~
T(u)
I::
dry weight
:I,---Pl_an_t--.-JI
thickstock valve
: :: ~w~m
Fig. 1. Block diagram of the transformed plant. The main uncertainty of the system is the variation of the retention level in the interval [0.3,0.8]. To get an accurate description of the effect of this uncertainty, the transformed system is simulated in open loop near the operating point, with PRBSs having the approximate amplitude of the benchmark step tests. This was performed for 25 equidistant retention levels. The inputoutput pairs were used to identify 25 linear 2 by 2 models, y(z) = pR(z)u(z) where :tIl is dry weight and Y2 is ash content, with a standard RLS identification scheme, (Ljung, 1991). In figures 2 and 3 Bode plots of the 25 identified models are shown, together with templates for some frequencies. Since the 25 identified model were
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~/O
The MIMO block diagram can be transformed into 4 equi valent SIM 0 loops. We first design 91 (z) i.e. the first loop. Since the system is a multivariable system, also 92 (z) has effect on this loop. It is however possible to estimate the effect of 92 with respect to the first loop with the frequency domain bounds for loop 2. Note that this is not equivalent to assuming that 92 = 0, and closing the system loop by loop. To achieve robust performance for the first loop we calculate Horowitz bounds that a nominal open loop must fulfill, and then design 91 (z) by manual loop shaping. The transfer function 91 (z)
92(Z) = 0.83z - 0.47 0.953z - 0.88 z- 1 z - 0.84 1.54z 2 - 1.375z + 0.673 0.642z 2 + 0.619z - 0.0227(4) Z2 - 0.227z + 0.064 Z2 - 0.124z + 0.114
In the lower half of figure 6 the bounds and the nominal open loop is shown. Notice again that the bounds are not completely satisfied. Given 92(Z) the prefilter Jz(z) is designed, so that the frequency response specifications are fulfilled, see figure 4: Jz(z) = 1.31z - 1.2 0.374z2 - 0.245z + 0.118
=
0.54z - 0.36 1.23z + 0.583 z - 1 z2 - 0.785z + 0.236 0.536z 2 + 0.517 z - 0.019 Z2 - 0.7852z + 0.236
1.lz 2 -
z - 0.886 (2)
was finally chosen. In the upper half of figure 6 the bounds and the nominal open loop is shown. Bounds and nominal system for loop 1
40
2°5~~;;;;;:::~~"".~15JJ~ . ~::;3~~~::::~~ -20r-======----::========""'............. o -350 -300 -250 -200 -150 -100 -50
Z2 -
Z
(5)
+ 0.253
It is not a serious shortcoming that the bounds are not completely satisfied, since the bounds emanate from the approximate frequency domain specification. The bounds give however an indication of what frequencies might contribute to difficulties in satisfying the timedomain specifications. Bode plots of the controllers are shown in figure 7.
•~t
M8!J1i1ude of 91 and
92 (dotted)
~--.--.-.-o-r---~-----.--.--.--........ . .. 10--3
10-2
[rad/sl
Phase of 91 and g2 (dottid)
Bounds and nominal system for loop 2
i
_~:i:=:::::::1~,=====:~--.--.-.-o-r---.",-,-,-::---... .
-------.---.--..---....:::1.... ..
10-2 [radls) Ma",itude of 11 and 12 (cloned)
10-3
10- 1
(radls]
Fig. 6. Horowitz bounds for specifications 1 and 2
Fig. 7. Bode plots of 91 and 92 and magnitude plots of JI and Jz
Notice that it is impossible to satisfy the Horowitz bound constraint, due to the delay in the plant that limits the bandwidth. Given 9dz) the prefilter JI(z) is designed, so that the frequency response specifications are approximately fulfilled, see figure 4: JI(z) = 2.9z - 2.39 0.412z2 - 0.368z + 0.18 z - 0.49 z2 - 0.848z + 0.0723
(3)
Given 91 (z) it can be included in the plant, and the remaining design is a SIMO problem. Horowitz bounds that the second loop must fulfill for a nominal case can now be calculated. Loop shaping of 92(Z) gives
The design must be checked in the time domain. Simulations for the linear equivalent systems in closed loop are given in figure 8. Notice that all specifications are satisfied. The achieved settling time for all retention cases are TB < 310 for specification 1 and Ts < 330 for specification 2. This is slower than some of the proposed designs in (Edlund, 1994), but none of them fulfills the overshoot specifications for all ret.entions! For specification 3 it can be noted that there is no oscillatory behavior for any of the retention levels and that the disturbance rejection is as good as any of the previously suggested controllers. Simulations for the non-linear plant controlled with G(s) and F(s), according to block diagram figure 5,
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