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Computer Aided Strategies for Tuning GPC Controllers: A Case Study
Copyright @ IFAC Control Systems Design, Bratislava, Slovak Republic, 2000
COMPUTER AIDED STRATEGIES FOR TUNING GPC CONTROLLERS: A CASE STUDY P. Tona *, J. Chebassier * and M. M'Saad **
* Laboratoire d'Automatique de Grenoble, INPG-CNRS, ENSIEG, B.P. 46, 38402 Saint Martin d'Heres, France, Paolino. Tona,Joel. [email protected] ** Laboratoire d'Automatique de Procede, ISMRA, 6 boulevard Marechal Juin, 14050 Caen, France, [email protected]
Abstract: Generalized predictive control, in its linear and analytical version, is a general-purpose technique that has been successfully applied to various classes of plants. However, when we move from stable and well-damped plants to plants with more complex dynamics (unstable, oscillatory), choosing the appropriate GPC parameters may become difficult. This paper introduces a case study which involves a rotary flexible joint, i.e. a system with an integrator and a stable lightly-damped mode, and shows that is indeed possible to tune quickly and effectively a GPC controller. This goal is achieved through a methodology based on the use of the usual sensitivity junctions as closed-loop performance indicators, on a simplified choice of the observer polynomial and on plant-oriented design guidelines. The proposed methodology finds an adequate support in SIMART, a CACSD package for advanced control. Copyright @20001FAC
Keywords: Generalized predictive control, Computer-aided control system design, Methodology, Sensitivity functions
1. INTRODUCTION
of design-relevant GPC parameters increase and it is difficult to find suitable tuning guidelines. In the following, a particular approach to GPC, called partial state reference model control (PSRMC), is considered (section 2). As for any other two-degrees-of-freedom linear control system, its performance and robustness are best analyzed in the frequency domain, by using the usual sensitivity functions (section 2.3). Design consists then in tuning the controller parameters in order to obtain suitable shapes of those sensitivity functions. This iterative process requires obviously a sound CACSD support, which is provided by the package SIMART (section 3).
Among leading advanced control techniques, predictive control has been enjoying a remarkable degree of acceptance both in laboratories and in industry. Several successful applications (see the impressive bibliography in (Clarke, 1996)) attest its flexibility, robustness to plant-model mismatch, ability to deal with time delays and nonminimum-phase behavior. Though a large number of those applications concerns process control, it should not be forgotten that standard GPC, as introduced by (Clarke et al., 1987), as well as its derived methods that have an analytic solution and yield a linear control system, remains a general purpose technique that can be applied to various classes of plants.
To find quickly an initial controller and to speed up the whole design process it is important to simplify the choice of the PSRMC parameters. In particular, the observer polynomial, which is the most difficult parameter to tune, can be specified using only two scalars (section 2.4.1). Information
However, it is true than when moving from stable and well damped plants to plants with more complex dynamics (unstable, oscillatory), the number
259
about the plant characteristics and dynamics can be effectively used, to tune those two parameters and the remaining ones (section 2.4).
I/
u (t)
vy(t) y(t)
To illustrate the feasibility of the proposed approach, a case study involving a rotary flexible link, that is, a system with an integrator and a stable lightly-damped mode, is reported (section 4).
I+--------{I: I/y
(t)
Fig. 1. Nominal control system
Wyd(q-l )eyf(t) = Wyn(q-l )ey(t) Wud(q-l )euf(t)
2. THE PSRMC APPROACH
= D(q-l )Wun(q-l )eu(t)
where:
2.1 Control model
• sh, ph and ch are respectively the starting, prediction and control horizons, • A (control weighting) is a positive scalar, • D(p) represents the disturbance model, • W (z-l) = D(Z-l)Wun~Z-l) and W (Z-l) =
The input-output behavior of the plant to be controlled is assumed to be approximated by the following backward-shift sampled data model
A(q-l)y(t) = B(q-l)U(t - d - 1)
u
+ vet) + wet)
Wud(Z
)
y
~::i:-:~ denote user-specified input and output frequency weightings, respectively.
D(q-l )v(t) = C(q-l het) where {u(t)} and {yet)} denote respectively the control variable and the measured plant output, {v(t)} and {w(t)} represent respectively the external disturbances and the involved modelling errors, {'"Y( t)} is a sequence of widely spread pulses of unknown magnitude or of independent random variables with zero mean values and finite variances, and d + 1 denotes the plant model total delay in sampling periods.
The underlying control law is obtained using the generalized predictive control approach proposed in (Clarke et al., 1987) and adopting a plant model re-parameterization which takes into account the frequency weightings and the disturbance model (M'Saad et al., 1990). Notice that to minimize the criterion of equation 1, eyf(t + j) is replaced by its best prediction in a minimum variance sense eyf(t + jlt), whose calculation requires the knowledge of the polynomial C(q-l), which is in turn part of the disturbance and noise model. Since C(q-l) is difficult to estimate, it is usually considered as a further design parameter, the predictor or observer dynamics.
2.2 Predictive control design
As shown in (M'Saad et al., 1990), the sequences {e u (t)} and {ey(t)} defined by
The PSRMC approach yields a linear controller that can be put in the RST form
ey(t) = yet) - ;3B(q-l)y*(t) eu(t) = u(t) - ;3A(q-l)y*(t + d + 1) with
suitable both to implementation and analysis. represent the input-output errors of the partial state reference model control which is mainly motivated by the ability to specify the tracking and regulation dynamics independently. The transfer function p*(z-l) = ~:~~=;~ denotes the desired partial state reference model providing the reference sequence {y*(t)} from a bounded set-point sequence {u*(t)}. ;3 is a scalar introduced to ensure a unit static tracking gain.
2.3 Control system performance The input-output behavior of the nominal control system shown in figure 1 can be described by
yet) = ;3B(q-l)y*(t) + S(q-l)vy(t) - T(q-l)vy(t)
+ pS(q-l )(vy(t) + vy(t))
The predictive control objective consists in minimizing the following frequency-weighted linearquadratic cost function in a receding-horizon sense
u(t) = ;3A(q-l)y*(t + d + 1)
+ S(q-l)vu(t) - T(q-l)vu(t) + RS(q-l)(vy(t) + vy(t))
(1)
subject to euf (t
+ i)
= 0 for ch ::; i
where the pulse transfer functions
< ph, with 260
shaping the usual sensitivity functions in search of an acceptable trade-off among often contradictory performance specifications. Looking at equation 2, it could be tempting to use a pole placement approach to obtain the desired closed-loop behavior. Unfortunately, this is not possible because Pf(q-1) depends on C(q-1), except for the four limit cases described in (Clarke and Mohtadi, 1989), in which the criterion parameters sh, ph, ch and >. can be used to specify directly Pf(q-1) (for instance, deadbeat control or internal model control can be obtained). Fig. 2. Nominal performances S(Z-I)
=
T(
-I)
=
nS(--I)
=
Z
'"
pS(Z-I)
=
However, some design guidelines can be found:
A(z-I)S(z-llD(,-I) Pc(z 1)
sh: the starting horizon is usually taken equal to d+ 1; ph: the prediction horizon is taken equal to the number of samples corresponding to the desired rise-time (that is, ph = tr/t s ); ch: the control horizon, which represents the degrees of freedom of the control u(t), can be taken equal to 1 for plants with simple dynamics (i.e. stable and well-damped) and should be increased otherwise (a good rule of thumb consists in adding to this default value, the number of "difficult", that is, unstable or close to instability, system poles) >.: the control weighting has a default value of zero (actually, it has to be a little bigger than the machine precision), and can be used to penalize the control action, for example, when trading performance for robustness (controller de- tuning).
z-d-IB(,-l)R(z-l) Pc(z 1) A(z-ljR(,-l l Pc (z 1) z-d-IB(z-ljS(z-ljD(,-l j Pd z I)
are the usual sensitivity functions characterizing the control system (figure 2). These functions, namely the sensitivity function S(z-l), the complementary sensitivity function T(z-l), the regulator times the sensitivity function RS(z-l) and the plant (model) times the sensitivity function PS(Z-l) have shown to be convenient quantifiers for nominal performance and stability robustness (M'Saad and Chebassier, 1996; Zhou et al., 1996). In order to meet design specifications their shapes need to be properly modeled, by adjusting design parameters within an iterative procedure. Nominal trackingdynamicsy(t) = /3B(z-l)y*(t) = /3p*(z-l )B(z-l )u*(t) are independent from regulation dynamics.
2.4.1. The observer dynamics Choosing the observer polynomial C(q-1) is probably the most difficult part in PSRMC (and GPC) design, with very few guidelines available. For stable and welldamped systems, (Yoon and Clarke, 1995) propose
Asymptotical stability of the control system depends on the characteristic polynomial (2)
which must be Hurwitz. The factorization underlines that its roots come both from the underlying receding horizon linear quadratic control Pf(q-1) (the feedback polynomial) and from C(q-1) (the state observer polynomial), in the spirit of the optimal estimation theory.
with
Q
in the vicinity of the slowest pole(s) of A.
Another method, valid for first-order systems with time delay, is suggested in (Normey-Rico et al., 1999). Worth citing is also the two-step design proposed by (Ansay and Wertz, 1997), which, however assume a different control structure than the one considered in this paper.
2.4 Choice of the design parameters
Some of the PSRMC design parameters are directly determined by performance requirements:
The approach considered here consists in using two scalars (observer horizon and damping) to project all the poles of A(q-1) inside a specific zone of the z-plane. Thus, all the observer poles can be placed at once, while preserving a relation with system dynamics. For instance, it is possible to obtain observer dynamics similar to those described by equation 3 (if the observer horizon corresponds to the slowest modes of A) or to obtain faster dynamics, if needed. Moreover, this method, which has been broadly inspired by the
• the reference generator P* (Z-l) specifying the dominant part of the tracking dynamics, • the internal disturbance model D(q-1) ensuring offset-free disturbance rejection; for instance, an integral action may be incorporated. The remaining ones are to be specified using the above mentioned iterative procedure, that is,
261
pole placement technique of (de Larminat, 1993) and is fully described in (Tona, 2000), helps to choose the observer dynamics even in the case of stable or oscillatory systems. 2.4.2. The frequency weightings If needed, design can be refined having recourse to the frequency weightings Wu(Z-l) and Wy(z-l), which allow a more accurate open-loop shaping. When dealing with complex dynamics, those extra degrees of freedom may prove necessary. For instance, in oscillatory system control, the input frequency weighting could be used to insert a notch filter, which is a possible alternative, though not a robust one, to active damping control.
Fig. 4. The rotary flexible joint beam track a desired command while minimizing resonance in the system.
3. THE CACSD PACKAGE SIMART 4.1 Control model
The SIMART project (M'Saad et al., 1997) has originated from the belief that if principles and techniques of advanced control are properly assimilated, simple and viable solutions to many control problems may be found with minor effort. SIMART has been conceived with the purpose of making these principles and techniques accessible and usable, presenting the state of the art for SISO control in a user-friendly menu-driven environment and bringing to the fore all the stages involved in any genuine control system design from control model identification to real-time experiment. SIMART proves invaluably helpful in supporting the aforementioned iterative design process (figure 3).
Some simple open-loop experiments show that the system has one integrator and an underdamped (~ ~ 0.4) oscillatory mode with natural frequency W n ~ 18rad/s. A sampling period of t s = 30ms has been chosen, which is reasonable from an open-loop point of view, since it yields about 10 samples per natural period, and is compatible with a closed-loop bandwidth (in terms of T(Z-l)) of up to ~ 21 rad/ s. Due to the presence of an integrator, a control model is best identified in closed loop. First, a PID controller, auto-tuned through a relay experiment, has been used to supply some closed-loop I/O sequences. Then, using the direct approach and the output error method, the following control model has been identified:
4. A CASE STUDY: A FLEXIBLE JOINT The system is a laboratory plant which is representative of the control problems encountered in large geared robot joints where flexibility is exhibited in the gearbox: a DC motor rotates a flexible joint to which is mounted one end of a rigid beam. Figure 4 shows the joint configuration that has been used for the experiments. The control objective consists in having the tip of the
with ao = 1, aj = -3.315803594, a2 = 4.199523705, a3 = -2.418411614, a1 = 0.5346915033,
bo = 0.005933163787, b1 = -0.003974530813,
d
= 1.
4.2 Control design
The control system is required to have a tracking bandwidth of WT- ~ 3 rad/ s (which corresponds to a rise time t r ~ 1 ms, including the discrete time delay) and for regulation. Other specifications are: • a modulus margin .6.M > 0.5 (> -6dB), • a delay margin .6.7 > 60ms (> 2ts).
c::J G ./ .'
'.
«
J t.
:7
All specifications but the tracking bandwidth can be directly translated into templates on the usual sensitivity functions, to be used as a visual landmarks for design. The templates depicted in figures 5 to 7 have been chosen considering that:
.,
Fig. 3. Support for iterative design
262
Complementary 5oeflSllf\llty
,0r------'----c~
'0r------'------,
'0
-'0
-'0
-20
-20
-20
-30
'0' Frequency
(~sl
'0'
'0'
6Or-----'---'---"
-<0'-----:------'7
'0'
'0'
100
Frequency (raQ'sl
Frequ&ncy (rad's)
F~(rad's)
SenSlfMfy"Planl
SeosrtMty"Reg.llalOr
SenSItNJtY"P\wlf
so,------'------,
'0
10
'0,----------,
'0
'" 30f-_......_ - .
.0
~ -10
il.
~
~ _10
:; 20
~ -20
]
10
~ °
-30
~
-20
-20
'0'
Frequency (~s)
'0'
.30L..LC--,0";;-·- - - - " 0 '
'0'
'0' Freq.»ncy(radfa)
FN
F... ~ncy(~s)
'0'
Fig. 6. Opening the loop at ws /2 (previous design dashed)
Fig. 5. In search of the observer horizon ha • additional control objectives are needed to obtain a better-balanced control system; • the requested control performance must be attainable.
Changing interactively in SIMART the value~ of the observer horizon ha (figure 5), a suitabl( initial design is obtained with ha = 5. The overal shapes of the usual sensitivity functions for thi~ first design can be considered acceptable with the exception of RS(Z-l), which is well aboY( odB, especially at high frequency. While not much can be done to change the low-frequency behavior of RS(Z-l), because of the integratOl and the feeble velocity constant of the plant (k v ~ 0.06), it is possible (and recommended) to improve its shape at high frequency. Moreover (not unexpectedly, since T(Z-1) and RS(z-1) arE linked) the complementary sensitivity T(Z-l) ha~ too little high-frequency roll-off. Indeed, this first control system has been designed to be faster than is necessary (as it is shown by the sensitivity function S(Z-l» because, in the following steps, bandwidth is to be traded for improvements in high-frequency roll-off of T(Z-l) and RS(Z-l).
For instance, to avoid an excessive control activity in the presence of high levels of output noise, the RS(Z-l) (the regulator times the sensitivity) function is required to be strictly proper, but its cutoff frequency must be chosen taking into consideration that, up to the controller bandwidth, this function goes approximately as P(Z-1) (actually, RS(Z-1) == T(z-1 )/P(Z-1 ».
4.2.1. Directly specified parameters To meet the tracking bandwidth specification WT', the reference generator P*(z-I), i.e. the filter on the reference, is taken equal to the discrete-time equivalent of a continuous fourth-order system induding two couples of coincident real poles in W1,2 = 6.28rad/s and W3,4 = 12.56rad/s. As for the disturbance model, even if an integrator is present in the plant, a further integral action is added in the controller to take care of input disturbances.
4.2.4. Improving the initial design Improvements can be obtained by acting on the control weighting ,\ (de-tuning) and/or by the introducing an appropriate frequency weighting. Although choosing frequency weightings is not al-
4.2.2. Initial criterion parameters The guidelines of section 2.4 suggest the following values for an initial design:
= d + 1 = 2, = tr/t s = 33, ch = 1 + number of "difficult"
• sh • ph
• • ,\ =
o.
poles
= 4,
'o'
Fr&que1lCY (rad/s)
so oo
~
4.2.3. Choosing the observer horizon Once all the other parameters have been selected, the observer polynomial C (q-1 ) is to be chosen, through the two scalars introduced in section 2.4.1, to get a suitable initial design. The observer damping having a lesser impact on design than the observer horizon, the former is kept fixed to its default value of 0.707, while varying the latter.
2
C
:r
:>
'0'
-<0
'0' Frequency (radls)
SenSltivItY'PIant
SensltMty"Re9Jlator
'0
30
~ -10·
20 10··
°
-'0 -20 -30
'0'
F~ncy(raQlsl
'0'
Fig. 7. De-tuning with ,\ (previous design dashed)
263
• a simplified choice of the observer polynomial; • a suitable CACSD support, through the SIMART package. 10
15
20
25
30
35
40
45
A case study involving a laboratory process with an integrator and oscillating dynamics has been reported, to show the possibilities offered by this approach.
SO
t!me(s)
6. REFERENCES Ansay, P. and V. Wertz (1997). Model uncertainties in GPC: a systematic two step design. In: Proceedings of the ECC97. Bruxelles, Belgium. Clarke, D. W. (1996). Recent advances in modelbased predictive control. In: Proc. COPERNICUS Spring School on Adaptive and Predictive Control. Oxford,UK. pp. 1-15. Clarke, D. W. and C. Mohtadi (1989). Properties of the generalized predictive control. Automatica 25(6), 859-875. Clarke, D. W., C. Mohtadi and P. S. Tuffs (1987). Generalized predictive control. A utomatica 23(2),137-160. de Larminat, Ph. (1993). Commande des systemes lineaires. Collection Automatique. first ed .. Hermes, France. M'Saad, M. and J. Chebassier (1996). Commande optimale. Chap. Commande predictive des systemes. Diderot Editeur, Arts et Sciences. M'Saad, M., I.D. Landau and M. Samman (1990). Further evaluation of partial state model adaptive design. Int. J. Adaptive Control and Signal Processing 4(2), 133-148. M'Saad, M., J. Chebassier and P. Tona (1997). Advanced control using the CACSD package SIMART. In: IFAC Conference on Systems, Structure and Control. Bucharest, Romania. pp. 384-389. Normey-Rico, J.E., C. Bordons and E.F. Camacho (1999). Simple prefilter design in GPC for a wide class of industrial processes. In: Proceedings of the 14th IFAC World Congress. Tona, P. (2000). Conception de systemes de commande avancee par calculateur: methodologie et applications. PhD thesis. Laboratoire d'Automatique de Grenoble, Institut National Polytechnique de Grenoble. Yoon, T. and D. Clarke (1995). Observer design in receding-horizon control. International Journal of Control 2, 171-19l. Zhou, K., J.C. Doyle and K. Glover (1996). Robust and Optimal Control. Prentice-Hall. Englewood Cliffs.
Fig. 8. Tracking and regulation performance ways straightforward, for this class of plants a simple option consists in "opening the loop" at the critical frequency w s /2, by using the input frequency weighting W u (z-1) = 1/(1 + z-I). As shown by figure 6, this choice improves the (very) high frequency behavior of RS(Z-1) and 7(z-1), whereas the initial performance is preserved inside the controller bandwidth. To force RS(Z-1) below its template, more drastic action needs to be taken, by means of a different frequency weighting or, more viably, by increasing A. Experiments with A (figure 7) show that its impact on the original design is considerable : a significant loss of bandwidth and higher peaks in S(z-l) and 7(Z-1) are observed. Nevertheless, increasing A is beneficial to the high-frequency roll-off of RS(Z-1) and 7(z-1), and a sensible trade-off can be obtained. The control design with A = 0.01 meets the specification and is held as final design. Results from controller implementation are shown in figure 8. Performance is fairly good even if the system is affected by several nonlinear phenomena (e.g. backlash, dead-zone). The oscillating behavior has been successfully damped for both tracking and regulation.
5. CONCLUDING REMARKS This paper has illustrated an approach for rapid tuning of generalized predictive controllers, applying not only to stable and well-damped systems but also to other classes of plants with more complex dynamics. The ingredients of this approach are: • t.he use of partial state reference model control, which allows a decoupling of tracking and regulation dynamics; • performance quantifying through the usual sensitivity functions; • relating design parameters to plant characteristics and dynamics;
264
COMPUTER AIDED STRATEGIES FOR TUNING GPC CONTROLLERS: A CASE STUDY P. Tona *, J. Chebassier * and M. M'Saad **
* Laboratoire d'Automatique de Grenoble, INPG-CNRS, ENSIEG, B.P. 46, 38402 Saint Martin d'Heres, France, Paolino. Tona,Joel. [email protected] ** Laboratoire d'Automatique de Procede, ISMRA, 6 boulevard Marechal Juin, 14050 Caen, France, [email protected]
Abstract: Generalized predictive control, in its linear and analytical version, is a general-purpose technique that has been successfully applied to various classes of plants. However, when we move from stable and well-damped plants to plants with more complex dynamics (unstable, oscillatory), choosing the appropriate GPC parameters may become difficult. This paper introduces a case study which involves a rotary flexible joint, i.e. a system with an integrator and a stable lightly-damped mode, and shows that is indeed possible to tune quickly and effectively a GPC controller. This goal is achieved through a methodology based on the use of the usual sensitivity junctions as closed-loop performance indicators, on a simplified choice of the observer polynomial and on plant-oriented design guidelines. The proposed methodology finds an adequate support in SIMART, a CACSD package for advanced control. Copyright @20001FAC
Keywords: Generalized predictive control, Computer-aided control system design, Methodology, Sensitivity functions
1. INTRODUCTION
of design-relevant GPC parameters increase and it is difficult to find suitable tuning guidelines. In the following, a particular approach to GPC, called partial state reference model control (PSRMC), is considered (section 2). As for any other two-degrees-of-freedom linear control system, its performance and robustness are best analyzed in the frequency domain, by using the usual sensitivity functions (section 2.3). Design consists then in tuning the controller parameters in order to obtain suitable shapes of those sensitivity functions. This iterative process requires obviously a sound CACSD support, which is provided by the package SIMART (section 3).
Among leading advanced control techniques, predictive control has been enjoying a remarkable degree of acceptance both in laboratories and in industry. Several successful applications (see the impressive bibliography in (Clarke, 1996)) attest its flexibility, robustness to plant-model mismatch, ability to deal with time delays and nonminimum-phase behavior. Though a large number of those applications concerns process control, it should not be forgotten that standard GPC, as introduced by (Clarke et al., 1987), as well as its derived methods that have an analytic solution and yield a linear control system, remains a general purpose technique that can be applied to various classes of plants.
To find quickly an initial controller and to speed up the whole design process it is important to simplify the choice of the PSRMC parameters. In particular, the observer polynomial, which is the most difficult parameter to tune, can be specified using only two scalars (section 2.4.1). Information
However, it is true than when moving from stable and well damped plants to plants with more complex dynamics (unstable, oscillatory), the number
259
about the plant characteristics and dynamics can be effectively used, to tune those two parameters and the remaining ones (section 2.4).
I/
u (t)
vy(t) y(t)
To illustrate the feasibility of the proposed approach, a case study involving a rotary flexible link, that is, a system with an integrator and a stable lightly-damped mode, is reported (section 4).
I+--------{I: I/y
(t)
Fig. 1. Nominal control system
Wyd(q-l )eyf(t) = Wyn(q-l )ey(t) Wud(q-l )euf(t)
2. THE PSRMC APPROACH
= D(q-l )Wun(q-l )eu(t)
where:
2.1 Control model
• sh, ph and ch are respectively the starting, prediction and control horizons, • A (control weighting) is a positive scalar, • D(p) represents the disturbance model, • W (z-l) = D(Z-l)Wun~Z-l) and W (Z-l) =
The input-output behavior of the plant to be controlled is assumed to be approximated by the following backward-shift sampled data model
A(q-l)y(t) = B(q-l)U(t - d - 1)
u
+ vet) + wet)
Wud(Z
)
y
~::i:-:~ denote user-specified input and output frequency weightings, respectively.
D(q-l )v(t) = C(q-l het) where {u(t)} and {yet)} denote respectively the control variable and the measured plant output, {v(t)} and {w(t)} represent respectively the external disturbances and the involved modelling errors, {'"Y( t)} is a sequence of widely spread pulses of unknown magnitude or of independent random variables with zero mean values and finite variances, and d + 1 denotes the plant model total delay in sampling periods.
The underlying control law is obtained using the generalized predictive control approach proposed in (Clarke et al., 1987) and adopting a plant model re-parameterization which takes into account the frequency weightings and the disturbance model (M'Saad et al., 1990). Notice that to minimize the criterion of equation 1, eyf(t + j) is replaced by its best prediction in a minimum variance sense eyf(t + jlt), whose calculation requires the knowledge of the polynomial C(q-l), which is in turn part of the disturbance and noise model. Since C(q-l) is difficult to estimate, it is usually considered as a further design parameter, the predictor or observer dynamics.
2.2 Predictive control design
As shown in (M'Saad et al., 1990), the sequences {e u (t)} and {ey(t)} defined by
The PSRMC approach yields a linear controller that can be put in the RST form
ey(t) = yet) - ;3B(q-l)y*(t) eu(t) = u(t) - ;3A(q-l)y*(t + d + 1) with
suitable both to implementation and analysis. represent the input-output errors of the partial state reference model control which is mainly motivated by the ability to specify the tracking and regulation dynamics independently. The transfer function p*(z-l) = ~:~~=;~ denotes the desired partial state reference model providing the reference sequence {y*(t)} from a bounded set-point sequence {u*(t)}. ;3 is a scalar introduced to ensure a unit static tracking gain.
2.3 Control system performance The input-output behavior of the nominal control system shown in figure 1 can be described by
yet) = ;3B(q-l)y*(t) + S(q-l)vy(t) - T(q-l)vy(t)
+ pS(q-l )(vy(t) + vy(t))
The predictive control objective consists in minimizing the following frequency-weighted linearquadratic cost function in a receding-horizon sense
u(t) = ;3A(q-l)y*(t + d + 1)
+ S(q-l)vu(t) - T(q-l)vu(t) + RS(q-l)(vy(t) + vy(t))
(1)
subject to euf (t
+ i)
= 0 for ch ::; i
where the pulse transfer functions
< ph, with 260
shaping the usual sensitivity functions in search of an acceptable trade-off among often contradictory performance specifications. Looking at equation 2, it could be tempting to use a pole placement approach to obtain the desired closed-loop behavior. Unfortunately, this is not possible because Pf(q-1) depends on C(q-1), except for the four limit cases described in (Clarke and Mohtadi, 1989), in which the criterion parameters sh, ph, ch and >. can be used to specify directly Pf(q-1) (for instance, deadbeat control or internal model control can be obtained). Fig. 2. Nominal performances S(Z-I)
=
T(
-I)
=
nS(--I)
=
Z
'"
pS(Z-I)
=
However, some design guidelines can be found:
A(z-I)S(z-llD(,-I) Pc(z 1)
sh: the starting horizon is usually taken equal to d+ 1; ph: the prediction horizon is taken equal to the number of samples corresponding to the desired rise-time (that is, ph = tr/t s ); ch: the control horizon, which represents the degrees of freedom of the control u(t), can be taken equal to 1 for plants with simple dynamics (i.e. stable and well-damped) and should be increased otherwise (a good rule of thumb consists in adding to this default value, the number of "difficult", that is, unstable or close to instability, system poles) >.: the control weighting has a default value of zero (actually, it has to be a little bigger than the machine precision), and can be used to penalize the control action, for example, when trading performance for robustness (controller de- tuning).
z-d-IB(,-l)R(z-l) Pc(z 1) A(z-ljR(,-l l Pc (z 1) z-d-IB(z-ljS(z-ljD(,-l j Pd z I)
are the usual sensitivity functions characterizing the control system (figure 2). These functions, namely the sensitivity function S(z-l), the complementary sensitivity function T(z-l), the regulator times the sensitivity function RS(z-l) and the plant (model) times the sensitivity function PS(Z-l) have shown to be convenient quantifiers for nominal performance and stability robustness (M'Saad and Chebassier, 1996; Zhou et al., 1996). In order to meet design specifications their shapes need to be properly modeled, by adjusting design parameters within an iterative procedure. Nominal trackingdynamicsy(t) = /3B(z-l)y*(t) = /3p*(z-l )B(z-l )u*(t) are independent from regulation dynamics.
2.4.1. The observer dynamics Choosing the observer polynomial C(q-1) is probably the most difficult part in PSRMC (and GPC) design, with very few guidelines available. For stable and welldamped systems, (Yoon and Clarke, 1995) propose
Asymptotical stability of the control system depends on the characteristic polynomial (2)
which must be Hurwitz. The factorization underlines that its roots come both from the underlying receding horizon linear quadratic control Pf(q-1) (the feedback polynomial) and from C(q-1) (the state observer polynomial), in the spirit of the optimal estimation theory.
with
Q
in the vicinity of the slowest pole(s) of A.
Another method, valid for first-order systems with time delay, is suggested in (Normey-Rico et al., 1999). Worth citing is also the two-step design proposed by (Ansay and Wertz, 1997), which, however assume a different control structure than the one considered in this paper.
2.4 Choice of the design parameters
Some of the PSRMC design parameters are directly determined by performance requirements:
The approach considered here consists in using two scalars (observer horizon and damping) to project all the poles of A(q-1) inside a specific zone of the z-plane. Thus, all the observer poles can be placed at once, while preserving a relation with system dynamics. For instance, it is possible to obtain observer dynamics similar to those described by equation 3 (if the observer horizon corresponds to the slowest modes of A) or to obtain faster dynamics, if needed. Moreover, this method, which has been broadly inspired by the
• the reference generator P* (Z-l) specifying the dominant part of the tracking dynamics, • the internal disturbance model D(q-1) ensuring offset-free disturbance rejection; for instance, an integral action may be incorporated. The remaining ones are to be specified using the above mentioned iterative procedure, that is,
261
pole placement technique of (de Larminat, 1993) and is fully described in (Tona, 2000), helps to choose the observer dynamics even in the case of stable or oscillatory systems. 2.4.2. The frequency weightings If needed, design can be refined having recourse to the frequency weightings Wu(Z-l) and Wy(z-l), which allow a more accurate open-loop shaping. When dealing with complex dynamics, those extra degrees of freedom may prove necessary. For instance, in oscillatory system control, the input frequency weighting could be used to insert a notch filter, which is a possible alternative, though not a robust one, to active damping control.
Fig. 4. The rotary flexible joint beam track a desired command while minimizing resonance in the system.
3. THE CACSD PACKAGE SIMART 4.1 Control model
The SIMART project (M'Saad et al., 1997) has originated from the belief that if principles and techniques of advanced control are properly assimilated, simple and viable solutions to many control problems may be found with minor effort. SIMART has been conceived with the purpose of making these principles and techniques accessible and usable, presenting the state of the art for SISO control in a user-friendly menu-driven environment and bringing to the fore all the stages involved in any genuine control system design from control model identification to real-time experiment. SIMART proves invaluably helpful in supporting the aforementioned iterative design process (figure 3).
Some simple open-loop experiments show that the system has one integrator and an underdamped (~ ~ 0.4) oscillatory mode with natural frequency W n ~ 18rad/s. A sampling period of t s = 30ms has been chosen, which is reasonable from an open-loop point of view, since it yields about 10 samples per natural period, and is compatible with a closed-loop bandwidth (in terms of T(Z-l)) of up to ~ 21 rad/ s. Due to the presence of an integrator, a control model is best identified in closed loop. First, a PID controller, auto-tuned through a relay experiment, has been used to supply some closed-loop I/O sequences. Then, using the direct approach and the output error method, the following control model has been identified:
4. A CASE STUDY: A FLEXIBLE JOINT The system is a laboratory plant which is representative of the control problems encountered in large geared robot joints where flexibility is exhibited in the gearbox: a DC motor rotates a flexible joint to which is mounted one end of a rigid beam. Figure 4 shows the joint configuration that has been used for the experiments. The control objective consists in having the tip of the
with ao = 1, aj = -3.315803594, a2 = 4.199523705, a3 = -2.418411614, a1 = 0.5346915033,
bo = 0.005933163787, b1 = -0.003974530813,
d
= 1.
4.2 Control design
The control system is required to have a tracking bandwidth of WT- ~ 3 rad/ s (which corresponds to a rise time t r ~ 1 ms, including the discrete time delay) and for regulation. Other specifications are: • a modulus margin .6.M > 0.5 (> -6dB), • a delay margin .6.7 > 60ms (> 2ts).
c::J G ./ .'
'.
«
J t.
:7
All specifications but the tracking bandwidth can be directly translated into templates on the usual sensitivity functions, to be used as a visual landmarks for design. The templates depicted in figures 5 to 7 have been chosen considering that:
.,
Fig. 3. Support for iterative design
262
Complementary 5oeflSllf\llty
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Fig. 6. Opening the loop at ws /2 (previous design dashed)
Fig. 5. In search of the observer horizon ha • additional control objectives are needed to obtain a better-balanced control system; • the requested control performance must be attainable.
Changing interactively in SIMART the value~ of the observer horizon ha (figure 5), a suitabl( initial design is obtained with ha = 5. The overal shapes of the usual sensitivity functions for thi~ first design can be considered acceptable with the exception of RS(Z-l), which is well aboY( odB, especially at high frequency. While not much can be done to change the low-frequency behavior of RS(Z-l), because of the integratOl and the feeble velocity constant of the plant (k v ~ 0.06), it is possible (and recommended) to improve its shape at high frequency. Moreover (not unexpectedly, since T(Z-1) and RS(z-1) arE linked) the complementary sensitivity T(Z-l) ha~ too little high-frequency roll-off. Indeed, this first control system has been designed to be faster than is necessary (as it is shown by the sensitivity function S(Z-l» because, in the following steps, bandwidth is to be traded for improvements in high-frequency roll-off of T(Z-l) and RS(Z-l).
For instance, to avoid an excessive control activity in the presence of high levels of output noise, the RS(Z-l) (the regulator times the sensitivity) function is required to be strictly proper, but its cutoff frequency must be chosen taking into consideration that, up to the controller bandwidth, this function goes approximately as P(Z-1) (actually, RS(Z-1) == T(z-1 )/P(Z-1 ».
4.2.1. Directly specified parameters To meet the tracking bandwidth specification WT', the reference generator P*(z-I), i.e. the filter on the reference, is taken equal to the discrete-time equivalent of a continuous fourth-order system induding two couples of coincident real poles in W1,2 = 6.28rad/s and W3,4 = 12.56rad/s. As for the disturbance model, even if an integrator is present in the plant, a further integral action is added in the controller to take care of input disturbances.
4.2.4. Improving the initial design Improvements can be obtained by acting on the control weighting ,\ (de-tuning) and/or by the introducing an appropriate frequency weighting. Although choosing frequency weightings is not al-
4.2.2. Initial criterion parameters The guidelines of section 2.4 suggest the following values for an initial design:
= d + 1 = 2, = tr/t s = 33, ch = 1 + number of "difficult"
• sh • ph
• • ,\ =
o.
poles
= 4,
'o'
Fr&que1lCY (rad/s)
so oo
~
4.2.3. Choosing the observer horizon Once all the other parameters have been selected, the observer polynomial C (q-1 ) is to be chosen, through the two scalars introduced in section 2.4.1, to get a suitable initial design. The observer damping having a lesser impact on design than the observer horizon, the former is kept fixed to its default value of 0.707, while varying the latter.
2
C
:r
:>
'0'
-<0
'0' Frequency (radls)
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SensltMty"Re9Jlator
'0
30
~ -10·
20 10··
°
-'0 -20 -30
'0'
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Fig. 7. De-tuning with ,\ (previous design dashed)
263
• a simplified choice of the observer polynomial; • a suitable CACSD support, through the SIMART package. 10
15
20
25
30
35
40
45
A case study involving a laboratory process with an integrator and oscillating dynamics has been reported, to show the possibilities offered by this approach.
SO
t!me(s)
6. REFERENCES Ansay, P. and V. Wertz (1997). Model uncertainties in GPC: a systematic two step design. In: Proceedings of the ECC97. Bruxelles, Belgium. Clarke, D. W. (1996). Recent advances in modelbased predictive control. In: Proc. COPERNICUS Spring School on Adaptive and Predictive Control. Oxford,UK. pp. 1-15. Clarke, D. W. and C. Mohtadi (1989). Properties of the generalized predictive control. Automatica 25(6), 859-875. Clarke, D. W., C. Mohtadi and P. S. Tuffs (1987). Generalized predictive control. A utomatica 23(2),137-160. de Larminat, Ph. (1993). Commande des systemes lineaires. Collection Automatique. first ed .. Hermes, France. M'Saad, M. and J. Chebassier (1996). Commande optimale. Chap. Commande predictive des systemes. Diderot Editeur, Arts et Sciences. M'Saad, M., I.D. Landau and M. Samman (1990). Further evaluation of partial state model adaptive design. Int. J. Adaptive Control and Signal Processing 4(2), 133-148. M'Saad, M., J. Chebassier and P. Tona (1997). Advanced control using the CACSD package SIMART. In: IFAC Conference on Systems, Structure and Control. Bucharest, Romania. pp. 384-389. Normey-Rico, J.E., C. Bordons and E.F. Camacho (1999). Simple prefilter design in GPC for a wide class of industrial processes. In: Proceedings of the 14th IFAC World Congress. Tona, P. (2000). Conception de systemes de commande avancee par calculateur: methodologie et applications. PhD thesis. Laboratoire d'Automatique de Grenoble, Institut National Polytechnique de Grenoble. Yoon, T. and D. Clarke (1995). Observer design in receding-horizon control. International Journal of Control 2, 171-19l. Zhou, K., J.C. Doyle and K. Glover (1996). Robust and Optimal Control. Prentice-Hall. Englewood Cliffs.
Fig. 8. Tracking and regulation performance ways straightforward, for this class of plants a simple option consists in "opening the loop" at the critical frequency w s /2, by using the input frequency weighting W u (z-1) = 1/(1 + z-I). As shown by figure 6, this choice improves the (very) high frequency behavior of RS(Z-1) and 7(z-1), whereas the initial performance is preserved inside the controller bandwidth. To force RS(Z-1) below its template, more drastic action needs to be taken, by means of a different frequency weighting or, more viably, by increasing A. Experiments with A (figure 7) show that its impact on the original design is considerable : a significant loss of bandwidth and higher peaks in S(z-l) and 7(Z-1) are observed. Nevertheless, increasing A is beneficial to the high-frequency roll-off of RS(Z-1) and 7(z-1), and a sensible trade-off can be obtained. The control design with A = 0.01 meets the specification and is held as final design. Results from controller implementation are shown in figure 8. Performance is fairly good even if the system is affected by several nonlinear phenomena (e.g. backlash, dead-zone). The oscillating behavior has been successfully damped for both tracking and regulation.
5. CONCLUDING REMARKS This paper has illustrated an approach for rapid tuning of generalized predictive controllers, applying not only to stable and well-damped systems but also to other classes of plants with more complex dynamics. The ingredients of this approach are: • t.he use of partial state reference model control, which allows a decoupling of tracking and regulation dynamics; • performance quantifying through the usual sensitivity functions; • relating design parameters to plant characteristics and dynamics;
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