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Control Under Output Inequality Constraints for Non Minimum Phase System
Copyright @ IFAC Management and Control of Production and Logistics, Grenoble, France, 2000
CONTROL UNDER OUTPUT INEQUALITY CONSTRAINTS FOR NON MINIMUM PHASE SYSTEM Guy Bornard, Costin Ene
Laboratoire d'Automatique de Grenoble, UMR CNRS 5528 ENSIEG-INPG, BP 46, 38402, St. Martin d'Heres, France
Abstract: The receding horizon optimal control techniques (predictive control) allow to cope easily with inequality constraints on inputs and outputs. However constraints on the outputs may generate instabilities when the system has unstable zeros. A "positive response" (PR) property is defined for the system. It is possible to design stable receding horizon control laws of systems having this property. A control strategy is also proposed for non-PR systems. Copyright @2000 [FAC Keywords: predictive control, constraints, non-minimum phase systems
1. INTRODUCTION
solution is proposed to ensure stability for NMP systems that are either PR or non PR. The paper is organized as follows. Preliminaries on receding control and input/output constraints defined in Section 2. In Section 3, the definition of a PR system and the related properties are given. The constraints problem on the output of a non-PR system and a stabilizing algorithm are treated in Section 4. Concluding remarks and future work end the paper.
This paper is concerned with model predictive control. The MPC approach covers a variety of methods using constrained receding horizon optimisation techniques. The basic aspects are to be found in (Bornard and Gauthier, 1983), (Prett and Garcia, 1988), (Keerthi and Gilbert, 1988), (Scokaert and Mayne, 1998). Thestabibility under constraints is discussed in (Gomez and Goodwin, 1996) and (Bemporad et al., 1997), (Alamir and Bornard, 1995), (Alamir and Marchand, 1999) for nonlinear systems. A special attention is paid to systems with unstable zeros in (Rawlings and Muske, 1993), (Horowitz et al., 1986). A geometrical insight to these systems can be found in (Commault and Dion, 1982), (Wonham, 1985).
2. RECEDING HORIZON CONSTRAINED CONTROL The frame of the work is the receding horizon quadratic optimisation. Receding horizon or moving window control can be considered as a limited memory control. The idea is to compute an optimal control law (that minimizes a certain quadratic criterion) in a finite number of steps. Thus, the memory of the controller is fixed. More precisely, consider the discrete-time system
In the present paper the authors examine the case of a linear non minimum phase time-invariant discrete system subject to output inequality constraints. Bad behaviour can be encountered while leaving an active inequality constraint. It is shown that such problems are linked to a subclass of the NMP systems which are not "positive response" systems. The property of being "positive response" (PR) is defined here and its analysis is presented. A finite receding horizon control
(1)
1081
where A E IR nxn , BE IR nxm , and C E IRl'xn are real, time-invariant matrices. The basic problem is to design a control law Uk in such a way that under suitable conditions the output Yk will track a given output ydk • At each sampling time, a criterion is minimised over the horizon, and the first element of the solution is applied.
while tracking the desired output generated by a delay plus first order reference model. An upper constraint is set on the output (Ymax = .8). The first step on the setpoint shows the unconstrained response. The second one leads to an active constraint. The third one drives the system back to the interior of the admissible domain. Everything works well since the output follows the desired reference whenever the constraint is not active.
Let us consider the linear quadratic (LQ) optimization cost function on a finite horizon N:
On the figure 2, the same protocol was applied to the system (4). The result obtained differs from the previous one by the end of the response : the control cannot drive the output towards the desired value because it would roughly need to violate the maximum value contraint. The behaviour of the system system could also exhibit numerical instabilities for the optimisation pro}}.. lem to be solved (non wellposedness, inexistence of a solution...). Details will be given in section 4.
k+N Jk
=
L
eTQei
+ ~uT R~Ui
(2)
i=k+l
= ydi - Yi and output error and, respectively, the input variation at each sample time. The following notations will be used : ~ 0, R > 0, where ei ~Ui = Ui-l - Ui-2 represent the
with Q
Uk.N
=
[..:U·
It should be noted that both systems (3) and (4) have unstable zeros. However only the second one behaves "badly". Eventhough the presence of unstable zeros has something to do with the problem exhibited here, this property is not sufficient to describe the situation. It will be shown that the good behaviour is determined by a "positive response" property, a notion that will be introduced in this paper.
idem for y, yd;
'1'=
This approach permits to include easily inequality constraints in the control strategy, since this leads to wellmastered techniques of convex quadratic optimisation under linear inequality constraints. The inequality constraints most generally encountered are those affecting the control inputs. Inequality constraints on outputs, Le. on linear combinations of states and inputs, can be easily integrated in the same optimisation formalism. Thus we shall consider inequality constraints on both input and output: Umin :S Uk,N :S max and Ymin :S Yk,N :S Ymax. Ymin/max and Umin/max are vectors of appropriate dimensions.
u
~.~--:':,---:!:----:':------;..---,'----~-.E=E"=~~
--
However, the presence of these constraints on the output may lead to instability and/or computational problems. This is in particular the case for certain systems having unstable zeros. The example 1 introduces the difficulties thus encountered.
Fig. 1. Constraint on output: the standard case
Example 1. Consider the following two 8180 systems with unitary steady state gain, which differ by their C matrix :
A= A=
[.g .~] ,B =
[:~] ,C =
[.g .~] ,B = [:~],
[1.9, -0.9]
C = [3, -2]
(3)
(4)
. --
The figure 1 shows the step response of the system (3) and the response of the closed loop system
............... ---.,-.
~...-::..
Fig. 2. Constraint on output: a more difficult case
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3. POSITIVE RESPONSE SYSTEMS
Proof: Let {xst, Ust, Ysd be a steady state and assume that the system is initialized at Xo = O.
Let us consider the linear time invariant system (1) which will be assumed throughout the paper as being controllable, observable and right invertible. Moreover, the mapping C restricted to the set of steady states is supposed injective (steady state controllability). Definition 1. Assume that for any target steady state {xst, Ust, Ysd, where {ust} and {Ysd are the input and the output associated to the steady state {xst}, there exists a discrete time N and a control sequence u([O, N - 1]) such that the following conditions are fulfilled:
Consider the reference Yr parameterized by k r > 0 as follows: _ {Yst. if k < kr Yrk Yst 1'f k > _ kr
(5)
Since the system (1) is right invertible, there exists k r > 0 and an input function U r such that the corresponding output matches exactly such reference Yr y(k,u,O)
= Yrk' k > 0
(6)
Remark that for systems with a relative degree vector r, it is possible to take kr = max(rd. The system (1) has its internal dynamics stable, then one has : lim(xk) = Xst (7) k -+ 00
i) Xst = x(N, u, 0) ii) Yst = y(N, u, 0) iii) sign(y(k,u,O)) = sign(Ystk) (elementwise) for k>O
As a consequence, for any € > 0 there exist k s such that IIxk - xstll < € for k ~ k s .
where x(k, u, ~o) and y(k, u, ~o) = Cx(k, u, ~o) are the solution of the system (1) at k for the input sequence u and the initialisation Xo = ~o.
Consider the input function Us parameterized by k s > 0 as follows : :
Then the system (1) is said to have a positive response, or be a "positive response" system (PR system).
U Sk
_ {urk if k < ks Ust 1'f k > _ ks
(8)
Since after the first feedback the system is zeromemory, one has Yk = Yst for every k > k s + n. Moreover, for any ~ > 0 there exist € such that IIxks - xstll < € implies IIYk - Ystll < ~ for every k ~ k s . Then for any ~ > 0 one can find k s such that IIYk - Yst 11 < ~ for every k ~ k s .
Remark that the system (3) uppermentionned is clearly a PR-system while this is presumably not the case for the system (4). Let us introduce here the main properties which characterize PR-systems.
Taking for instance ~ = mini=l .... .P(!Ysti 1}/2 one obtains that : Yk = Yrk for k ~ k s IIYk - Ystll < . min (IYSti 1}/2 for k > k s 1=1,... ,p
Proposition 1. The property of having a positive response is invariant with respect to the state feedback transformation group.
Then sign(y(k, Us, 0)) = sign(Ystk)' Vk Proof: Assume that (1) is positive response and consider a system (1') obtained from (1) through a state feedback defined by u = v + Kx. Then, for each trajectory {il, X, y} of the system (1), the corresponding input fj = il - Kx gives a trajectory of the system (1 ') with exactly the same x and y when both systems are equally initialized. Then if the condition "there exists u([O, N]) such that..." in the definition 1 applies to system (1), it applies also to system (1'). Since the state feedbacks constitute a group of transformations, the converse is true. 0
~
0
(9) 0
Proposition 3. Assume that the system (1) is a PR system, then for each output, at least one of the corresponding step responses has its first nonzero element of the same sign as the steady state gain: Vi E {1, ... p},3j E {l, ... m} such that sign(R;jr) = sign(Gij ), where R;jk is the kth element of the step response, G is the steady .)
ij
state gain and rij is the relative degree of the transfer between the input j and the output i of the system (1). The proof of this proposition is left to the reader.
From the proposition 1, it could be sufficient to restrict our attention, at first, to nilpotent systems.
Propositions 2 and 3 provide only a link between positive response systems and systems with unstable zeros, and not a necessary and sufficient condition. In fact, positive response systems with unstable zeros can be found, as shown by following example.
Proposition 2. Assume that all the zeros of the system (1) are stable. Then (1) is a positive response system.
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The figure 7 shows that the system is now able to follow the desired reference. The counterpart is that when the constraint is active for a long time, there is a gap between the maximum admissible value and the actual maximum value of the output. The following interesting features should be pointed out : • The uppermentionned gap cannot be avoided • The constraint is satisfied during the transient time as well as the steady state by minimizing the criterion, without any kind of sequential supervision of the threshold • The tuning is straightforward since the parameters are a direct image of the performances to be reached.
i
~
i
..
L
f .. ~
r i
u
...
-
,-~
- .... ....
- M _ .......
-.,.....
'
Fig. 7. Modified constraint: k esc = 5, dYesc = .1
5. CONCLUSION In this paper we discussed the case of receding horizon predictive control of time-invariant SISO or MIMO linear systems when the output is subject to inequality constraints. The related new concepts of positive response system and of bounded time positive response system were introduced. An algorithm is provided for treating the case of the nonPR systems. The settings of the algorithm are made through parameters having a straightforward meaning in terms of performance. An important feature not treated here is concerning the management of the unknown disturbances while matching output constraints. The ideas presented in this paper could be as well used for extending usual methods (set range, funnel) to the case of non positive response systems.
6. REFERENCES Alamir, M. and G. Bornard (1995). Stability of a truncated infinite constrained receding horizon scheme: the general discrete nonlinear case. Automatica 31(9),1353-1356.
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Alamir, M. and N. Marchand (1999). Numerical stabilisation of nonlinear systems: exact theory and approximate numerical implementation. European Journal of Control 5(1), 8797. Bai, E-W. and S. Dasgupta (1996). A minimal k-step delay controller for robust tracking of non-minimum phase systems. Systems f.1 Control Letters 28, 197-203. Bemporad, A., A. Casavola and E. Mosca (1997). Nonlinear control of constrained linear systems via predicitve reference management. IEEE Trans. on Automatic Control 42(32), 340-349. Bornard, G. and J.-P. Gauthier (1983). Commande multivariable en presence de contraintes de type inegalit. RAIRO, Autom. Syst. Anal. Control 17, 205-222. Commault, C. and J.-M. Dion (1982). structure at infinity of linear multivariable systems: a geometric approach. IEEE Trans. on Automatic Control 27, 693--696. Gomez, G. I. and G. C. Goodwin (1996). Integral constraints on sensitivity vectors for multivariable linear systems. Automatica 32(4),499-518. Horowitz, I. M., S. Oldak and O. Yaniv (1986). An important property of non-minimumphase multiple-input-multiple-output feedback systems. International Journal of Control 44(3), 677--688. Keerthi, S. S. and E. G. Gilbert (1988). Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations. Journal of Optimization Theory and Applications 57(2), 265-293. Prett, D. M. and C. E. Garcia (1988). Fundamental Process Control. Butterworths. Boston. Rawlings, J. B. and K. R. Muske (1993). Stability of constrained receding horizon control. IEEE Trans. on Automatic Control 38(10), 1512--6. Scokaert, P. O. M. and D. Q. Mayne (1998). Minmax model predictive control for constrained linear systems. IEEE Trans. on Automatic Control 43(8), 1136-1142. Wonham, W. M. (1985). Linear Multivariable Control - A Geometric Approach. SpringerVerlag. New York.
Copyright
OPTIMAL TRACKING RULE FOR AN ARMAX MODEL AND ITS USAGE FOR SUPERVISION Rudolf Razek .'•• , Jan Stecha·, Joseph Aguilar-Martin··
• 1rnka laboratory for automatic control, Czuh Tuhnical university, department of control, Tuhnicka 2, 166 27 Praha 6, Czuh Republic, Fax: +420-2-290 159, Tel. +420-2-2435 7345, e-mail: razektlfel.cvut.cz •• LAAS-CNRS, 7, av. du Colonel Roche, 310 77 Toulouse, l'rance
Abstract: In this contribution a cautious stochastic optimal tracking rule is derived and its properties studied. Obtained tracking rule has got the form of a state feedback. A supervision technique, based on a set of Kalman-like filters, for systems whose parameters are unknown, but bounded by intervals is shown. Copyright @ 2000 IFAC Keywords: Optimal control, Stochastic system, Uncertainty, Supervision
1. INTRODUCTION
a simultaneous state and parameter estimation can be performed. A detailed treatment of this problem can be found in (Salut, 1976). In this work it is shown that considering the noise parameters c; (see the model equation (1) bellow) to be known, the estimation procedure can be performed with no approximation involved, unlike other types of models where the estimation of extended state vector would lead to a nonlinear problem. Let us remark, that in (Salut, 1976) methods permitting to estimate the noise coefficients from measured data spectrum can be found as well. Finally, let us remark that there are many methods for identifying the ARMAX model order from input-output data. Thus, we can see that the chosen ARMAX model is rather suitable for our control problem. Moreover, the aim of this paper is not to estimate state and parameters of an unknown plant, but to control and to supervise it. That is why the estimation task will be described very briefly.
Let us suppose to have an unknown plant and a suitable method for identifying it. The task is to conceive a regulator suitable to take into account all the information delivered by the identification process, but not necessarily a control that would improve the estimation. The usual way in case of stochastic systems (state is a random variable) with unknown or not precisely known parameters described by a stochastic distribution is to use a controller based on the certainty equivalent (CE) principle. This means that only the estimated mean of unknown parameters is used. They are performing well, but our idea was to find a control/tracking rule that would be optimal for a large family of systems that are not very "far away" from a nominal controller. Thus, we are considering a family of systems given by a normal distribution p(8) '" N where 8 is the vector of
(n,p/I)
n
/I
parameters of the model. Both moments and P are used for determining the control rule. This means that all the possible knowledge of the estimated system is exploited. That is why we call our strategy cautious.
The paper is organized in the following way. The first part deals with the problem of stochastic optimal tracking. The second part shows how this optimal tracking can be used for supervision purposes.
In our approach a linear ARMAX model of known order was chosen because it is easy to handle and it can be used as a model for a large family of linear stochastic systems. Moreover, in the literature it was shown that
The first part is split into three sections. In the first section, the ARMAX model is introduced and the problem of simultaneous state and parameter estimation is treated. In the second section, the problem of stochastic 1087
The figure 7 shows that the system is now able to follow the desired reference. The counterpart is that when the constraint is active for a long time, there is a gap between the maximum admissible value and the actual maximum value of the output. The following interesting features should be pointed out: • The uppermentionned gap cannot be avoided • The constraint is satisfied during the transient time as well as the steady state by minimizing the criterion, without any kind of sequential supervision of the threshold • The tuning is straightforward since the parameters are a direct image of the performances to be reached.
· · ..f _ •• CIpO.. ,
- . - 1 _.........
-., -.,
Fig. 7. Modified constraint: k esc
..,... _"",
= 5, dYesc = .1
5. CONCLUSION In this paper we discussed the case of receding horizon predictive control of time-invariant SISO or MIMO linear systems when the output is subject to inequality constraints. The related new concepts of positive response system and of bounded time positive response system were introduced. An algorithm is provided for treating the case of the nonPR systems. The settings of the algorithm are made through parameters having a straightforward meaning in terms of performance. An important feature not treated here is concerning the management of the unknown disturbances while matching output constraints. The ideas presented in this paper could be as well used for extending usual methods (set range, funnel) to the case of non positive response systems.
6. REFERENCES Alamir, M. and G. Bornard (1995). Stability of a truncated infinite constrained receding horizon scheme: the general discrete nonlinear case. Automatica 31(9),1353-1356.
1086
Alamir, M. and N. Marchand (1999). Numerical stabilisation of nonlinear systems: exact theory and approximate numerical implementation. European Journal of Control 5(1), 8797. Bai, E-W. and S. Dasgupta (1996). A minimal k-step delay controller for robust tracking of non-minimum phase systems. Systems €1 Control Letters 28, 197-203. Bemporad, A., A. Casavola and E. Mosca (1997). Nonlinear control of constrained linear systems via predicitve reference management. IEEE Trans. on Automatic Control 42(32), 340-349. Bornard, G. and J.-P. Gauthier (1983). Commande multivariable en presence de contraintes de type inegalit. RAIRO, Autom. Syst. Anal. Control 17, 205-222. Commault, C. and J.-M. Dion (1982). structure at infinity of linear multivariable systems: a geometric approach. IEEE Trans. on Automatic Control 27, 693--696. Gomez, G. I. and G. C. Goodwin (1996). Integral constraints on sensitivity vectors for multivariable linear systems. Automatica 32(4),499-518. Horowitz, I. M., S. Oldak and O. Yaniv (1986). An important property of non-minimumphase multiple-input-multiple-output feedback systems. International Journal of Control 44(3), 677-688. Keerthi, S. S. and E. G. Gilbert (1988). Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations. Journal of Optimization Theory and Applications 57(2), 265-293. Prett, D. M. and C. E. Garcia (1988). Fundamental Process Control. Butterworths. Boston. Rawlings, J. B. and K. R. Muske (1993). Stability of constrained receding horizon control. IEEE Trans. on Automatic Control 38(10), 1512--6. Scokaert, P. O. M. and D. Q. Mayne (1998). Minmax model predictive control for constrained linear systems. IEEE Trans. on Automatic Control 43(8), 1136-1142. Wonham, W. M. (1985). Linear Multivariable Control - A Geometric Approach. SpringerVerlag. New York.
CONTROL UNDER OUTPUT INEQUALITY CONSTRAINTS FOR NON MINIMUM PHASE SYSTEM Guy Bornard, Costin Ene
Laboratoire d'Automatique de Grenoble, UMR CNRS 5528 ENSIEG-INPG, BP 46, 38402, St. Martin d'Heres, France
Abstract: The receding horizon optimal control techniques (predictive control) allow to cope easily with inequality constraints on inputs and outputs. However constraints on the outputs may generate instabilities when the system has unstable zeros. A "positive response" (PR) property is defined for the system. It is possible to design stable receding horizon control laws of systems having this property. A control strategy is also proposed for non-PR systems. Copyright @2000 [FAC Keywords: predictive control, constraints, non-minimum phase systems
1. INTRODUCTION
solution is proposed to ensure stability for NMP systems that are either PR or non PR. The paper is organized as follows. Preliminaries on receding control and input/output constraints defined in Section 2. In Section 3, the definition of a PR system and the related properties are given. The constraints problem on the output of a non-PR system and a stabilizing algorithm are treated in Section 4. Concluding remarks and future work end the paper.
This paper is concerned with model predictive control. The MPC approach covers a variety of methods using constrained receding horizon optimisation techniques. The basic aspects are to be found in (Bornard and Gauthier, 1983), (Prett and Garcia, 1988), (Keerthi and Gilbert, 1988), (Scokaert and Mayne, 1998). Thestabibility under constraints is discussed in (Gomez and Goodwin, 1996) and (Bemporad et al., 1997), (Alamir and Bornard, 1995), (Alamir and Marchand, 1999) for nonlinear systems. A special attention is paid to systems with unstable zeros in (Rawlings and Muske, 1993), (Horowitz et al., 1986). A geometrical insight to these systems can be found in (Commault and Dion, 1982), (Wonham, 1985).
2. RECEDING HORIZON CONSTRAINED CONTROL The frame of the work is the receding horizon quadratic optimisation. Receding horizon or moving window control can be considered as a limited memory control. The idea is to compute an optimal control law (that minimizes a certain quadratic criterion) in a finite number of steps. Thus, the memory of the controller is fixed. More precisely, consider the discrete-time system
In the present paper the authors examine the case of a linear non minimum phase time-invariant discrete system subject to output inequality constraints. Bad behaviour can be encountered while leaving an active inequality constraint. It is shown that such problems are linked to a subclass of the NMP systems which are not "positive response" systems. The property of being "positive response" (PR) is defined here and its analysis is presented. A finite receding horizon control
(1)
1081
where A E IR nxn , BE IR nxm , and C E IRl'xn are real, time-invariant matrices. The basic problem is to design a control law Uk in such a way that under suitable conditions the output Yk will track a given output ydk • At each sampling time, a criterion is minimised over the horizon, and the first element of the solution is applied.
while tracking the desired output generated by a delay plus first order reference model. An upper constraint is set on the output (Ymax = .8). The first step on the setpoint shows the unconstrained response. The second one leads to an active constraint. The third one drives the system back to the interior of the admissible domain. Everything works well since the output follows the desired reference whenever the constraint is not active.
Let us consider the linear quadratic (LQ) optimization cost function on a finite horizon N:
On the figure 2, the same protocol was applied to the system (4). The result obtained differs from the previous one by the end of the response : the control cannot drive the output towards the desired value because it would roughly need to violate the maximum value contraint. The behaviour of the system system could also exhibit numerical instabilities for the optimisation pro}}.. lem to be solved (non wellposedness, inexistence of a solution...). Details will be given in section 4.
k+N Jk
=
L
eTQei
+ ~uT R~Ui
(2)
i=k+l
= ydi - Yi and output error and, respectively, the input variation at each sample time. The following notations will be used : ~ 0, R > 0, where ei ~Ui = Ui-l - Ui-2 represent the
with Q
Uk.N
=
[..:U·
It should be noted that both systems (3) and (4) have unstable zeros. However only the second one behaves "badly". Eventhough the presence of unstable zeros has something to do with the problem exhibited here, this property is not sufficient to describe the situation. It will be shown that the good behaviour is determined by a "positive response" property, a notion that will be introduced in this paper.
idem for y, yd;
'1'=
This approach permits to include easily inequality constraints in the control strategy, since this leads to wellmastered techniques of convex quadratic optimisation under linear inequality constraints. The inequality constraints most generally encountered are those affecting the control inputs. Inequality constraints on outputs, Le. on linear combinations of states and inputs, can be easily integrated in the same optimisation formalism. Thus we shall consider inequality constraints on both input and output: Umin :S Uk,N :S max and Ymin :S Yk,N :S Ymax. Ymin/max and Umin/max are vectors of appropriate dimensions.
u
~.~--:':,---:!:----:':------;..---,'----~-.E=E"=~~
--
However, the presence of these constraints on the output may lead to instability and/or computational problems. This is in particular the case for certain systems having unstable zeros. The example 1 introduces the difficulties thus encountered.
Fig. 1. Constraint on output: the standard case
Example 1. Consider the following two 8180 systems with unitary steady state gain, which differ by their C matrix :
A= A=
[.g .~] ,B =
[:~] ,C =
[.g .~] ,B = [:~],
[1.9, -0.9]
C = [3, -2]
(3)
(4)
. --
The figure 1 shows the step response of the system (3) and the response of the closed loop system
............... ---.,-.
~...-::..
Fig. 2. Constraint on output: a more difficult case
1082
3. POSITIVE RESPONSE SYSTEMS
Proof: Let {xst, Ust, Ysd be a steady state and assume that the system is initialized at Xo = O.
Let us consider the linear time invariant system (1) which will be assumed throughout the paper as being controllable, observable and right invertible. Moreover, the mapping C restricted to the set of steady states is supposed injective (steady state controllability). Definition 1. Assume that for any target steady state {xst, Ust, Ysd, where {ust} and {Ysd are the input and the output associated to the steady state {xst}, there exists a discrete time N and a control sequence u([O, N - 1]) such that the following conditions are fulfilled:
Consider the reference Yr parameterized by k r > 0 as follows: _ {Yst. if k < kr Yrk Yst 1'f k > _ kr
(5)
Since the system (1) is right invertible, there exists k r > 0 and an input function U r such that the corresponding output matches exactly such reference Yr y(k,u,O)
= Yrk' k > 0
(6)
Remark that for systems with a relative degree vector r, it is possible to take kr = max(rd. The system (1) has its internal dynamics stable, then one has : lim(xk) = Xst (7) k -+ 00
i) Xst = x(N, u, 0) ii) Yst = y(N, u, 0) iii) sign(y(k,u,O)) = sign(Ystk) (elementwise) for k>O
As a consequence, for any € > 0 there exist k s such that IIxk - xstll < € for k ~ k s .
where x(k, u, ~o) and y(k, u, ~o) = Cx(k, u, ~o) are the solution of the system (1) at k for the input sequence u and the initialisation Xo = ~o.
Consider the input function Us parameterized by k s > 0 as follows : :
Then the system (1) is said to have a positive response, or be a "positive response" system (PR system).
U Sk
_ {urk if k < ks Ust 1'f k > _ ks
(8)
Since after the first feedback the system is zeromemory, one has Yk = Yst for every k > k s + n. Moreover, for any ~ > 0 there exist € such that IIxks - xstll < € implies IIYk - Ystll < ~ for every k ~ k s . Then for any ~ > 0 one can find k s such that IIYk - Yst 11 < ~ for every k ~ k s .
Remark that the system (3) uppermentionned is clearly a PR-system while this is presumably not the case for the system (4). Let us introduce here the main properties which characterize PR-systems.
Taking for instance ~ = mini=l .... .P(!Ysti 1}/2 one obtains that : Yk = Yrk for k ~ k s IIYk - Ystll < . min (IYSti 1}/2 for k > k s 1=1,... ,p
Proposition 1. The property of having a positive response is invariant with respect to the state feedback transformation group.
Then sign(y(k, Us, 0)) = sign(Ystk)' Vk Proof: Assume that (1) is positive response and consider a system (1') obtained from (1) through a state feedback defined by u = v + Kx. Then, for each trajectory {il, X, y} of the system (1), the corresponding input fj = il - Kx gives a trajectory of the system (1 ') with exactly the same x and y when both systems are equally initialized. Then if the condition "there exists u([O, N]) such that..." in the definition 1 applies to system (1), it applies also to system (1'). Since the state feedbacks constitute a group of transformations, the converse is true. 0
~
0
(9) 0
Proposition 3. Assume that the system (1) is a PR system, then for each output, at least one of the corresponding step responses has its first nonzero element of the same sign as the steady state gain: Vi E {1, ... p},3j E {l, ... m} such that sign(R;jr) = sign(Gij ), where R;jk is the kth element of the step response, G is the steady .)
ij
state gain and rij is the relative degree of the transfer between the input j and the output i of the system (1). The proof of this proposition is left to the reader.
From the proposition 1, it could be sufficient to restrict our attention, at first, to nilpotent systems.
Propositions 2 and 3 provide only a link between positive response systems and systems with unstable zeros, and not a necessary and sufficient condition. In fact, positive response systems with unstable zeros can be found, as shown by following example.
Proposition 2. Assume that all the zeros of the system (1) are stable. Then (1) is a positive response system.
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The figure 7 shows that the system is now able to follow the desired reference. The counterpart is that when the constraint is active for a long time, there is a gap between the maximum admissible value and the actual maximum value of the output. The following interesting features should be pointed out : • The uppermentionned gap cannot be avoided • The constraint is satisfied during the transient time as well as the steady state by minimizing the criterion, without any kind of sequential supervision of the threshold • The tuning is straightforward since the parameters are a direct image of the performances to be reached.
i
~
i
..
L
f .. ~
r i
u
...
-
,-~
- .... ....
- M _ .......
-.,.....
'
Fig. 7. Modified constraint: k esc = 5, dYesc = .1
5. CONCLUSION In this paper we discussed the case of receding horizon predictive control of time-invariant SISO or MIMO linear systems when the output is subject to inequality constraints. The related new concepts of positive response system and of bounded time positive response system were introduced. An algorithm is provided for treating the case of the nonPR systems. The settings of the algorithm are made through parameters having a straightforward meaning in terms of performance. An important feature not treated here is concerning the management of the unknown disturbances while matching output constraints. The ideas presented in this paper could be as well used for extending usual methods (set range, funnel) to the case of non positive response systems.
6. REFERENCES Alamir, M. and G. Bornard (1995). Stability of a truncated infinite constrained receding horizon scheme: the general discrete nonlinear case. Automatica 31(9),1353-1356.
1086
Alamir, M. and N. Marchand (1999). Numerical stabilisation of nonlinear systems: exact theory and approximate numerical implementation. European Journal of Control 5(1), 8797. Bai, E-W. and S. Dasgupta (1996). A minimal k-step delay controller for robust tracking of non-minimum phase systems. Systems f.1 Control Letters 28, 197-203. Bemporad, A., A. Casavola and E. Mosca (1997). Nonlinear control of constrained linear systems via predicitve reference management. IEEE Trans. on Automatic Control 42(32), 340-349. Bornard, G. and J.-P. Gauthier (1983). Commande multivariable en presence de contraintes de type inegalit. RAIRO, Autom. Syst. Anal. Control 17, 205-222. Commault, C. and J.-M. Dion (1982). structure at infinity of linear multivariable systems: a geometric approach. IEEE Trans. on Automatic Control 27, 693--696. Gomez, G. I. and G. C. Goodwin (1996). Integral constraints on sensitivity vectors for multivariable linear systems. Automatica 32(4),499-518. Horowitz, I. M., S. Oldak and O. Yaniv (1986). An important property of non-minimumphase multiple-input-multiple-output feedback systems. International Journal of Control 44(3), 677--688. Keerthi, S. S. and E. G. Gilbert (1988). Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations. Journal of Optimization Theory and Applications 57(2), 265-293. Prett, D. M. and C. E. Garcia (1988). Fundamental Process Control. Butterworths. Boston. Rawlings, J. B. and K. R. Muske (1993). Stability of constrained receding horizon control. IEEE Trans. on Automatic Control 38(10), 1512--6. Scokaert, P. O. M. and D. Q. Mayne (1998). Minmax model predictive control for constrained linear systems. IEEE Trans. on Automatic Control 43(8), 1136-1142. Wonham, W. M. (1985). Linear Multivariable Control - A Geometric Approach. SpringerVerlag. New York.
Copyright
OPTIMAL TRACKING RULE FOR AN ARMAX MODEL AND ITS USAGE FOR SUPERVISION Rudolf Razek .'•• , Jan Stecha·, Joseph Aguilar-Martin··
• 1rnka laboratory for automatic control, Czuh Tuhnical university, department of control, Tuhnicka 2, 166 27 Praha 6, Czuh Republic, Fax: +420-2-290 159, Tel. +420-2-2435 7345, e-mail: razektlfel.cvut.cz •• LAAS-CNRS, 7, av. du Colonel Roche, 310 77 Toulouse, l'rance
Abstract: In this contribution a cautious stochastic optimal tracking rule is derived and its properties studied. Obtained tracking rule has got the form of a state feedback. A supervision technique, based on a set of Kalman-like filters, for systems whose parameters are unknown, but bounded by intervals is shown. Copyright @ 2000 IFAC Keywords: Optimal control, Stochastic system, Uncertainty, Supervision
1. INTRODUCTION
a simultaneous state and parameter estimation can be performed. A detailed treatment of this problem can be found in (Salut, 1976). In this work it is shown that considering the noise parameters c; (see the model equation (1) bellow) to be known, the estimation procedure can be performed with no approximation involved, unlike other types of models where the estimation of extended state vector would lead to a nonlinear problem. Let us remark, that in (Salut, 1976) methods permitting to estimate the noise coefficients from measured data spectrum can be found as well. Finally, let us remark that there are many methods for identifying the ARMAX model order from input-output data. Thus, we can see that the chosen ARMAX model is rather suitable for our control problem. Moreover, the aim of this paper is not to estimate state and parameters of an unknown plant, but to control and to supervise it. That is why the estimation task will be described very briefly.
Let us suppose to have an unknown plant and a suitable method for identifying it. The task is to conceive a regulator suitable to take into account all the information delivered by the identification process, but not necessarily a control that would improve the estimation. The usual way in case of stochastic systems (state is a random variable) with unknown or not precisely known parameters described by a stochastic distribution is to use a controller based on the certainty equivalent (CE) principle. This means that only the estimated mean of unknown parameters is used. They are performing well, but our idea was to find a control/tracking rule that would be optimal for a large family of systems that are not very "far away" from a nominal controller. Thus, we are considering a family of systems given by a normal distribution p(8) '" N where 8 is the vector of
(n,p/I)
n
/I
parameters of the model. Both moments and P are used for determining the control rule. This means that all the possible knowledge of the estimated system is exploited. That is why we call our strategy cautious.
The paper is organized in the following way. The first part deals with the problem of stochastic optimal tracking. The second part shows how this optimal tracking can be used for supervision purposes.
In our approach a linear ARMAX model of known order was chosen because it is easy to handle and it can be used as a model for a large family of linear stochastic systems. Moreover, in the literature it was shown that
The first part is split into three sections. In the first section, the ARMAX model is introduced and the problem of simultaneous state and parameter estimation is treated. In the second section, the problem of stochastic 1087
The figure 7 shows that the system is now able to follow the desired reference. The counterpart is that when the constraint is active for a long time, there is a gap between the maximum admissible value and the actual maximum value of the output. The following interesting features should be pointed out: • The uppermentionned gap cannot be avoided • The constraint is satisfied during the transient time as well as the steady state by minimizing the criterion, without any kind of sequential supervision of the threshold • The tuning is straightforward since the parameters are a direct image of the performances to be reached.
· · ..f _ •• CIpO.. ,
- . - 1 _.........
-., -.,
Fig. 7. Modified constraint: k esc
..,... _"",
= 5, dYesc = .1
5. CONCLUSION In this paper we discussed the case of receding horizon predictive control of time-invariant SISO or MIMO linear systems when the output is subject to inequality constraints. The related new concepts of positive response system and of bounded time positive response system were introduced. An algorithm is provided for treating the case of the nonPR systems. The settings of the algorithm are made through parameters having a straightforward meaning in terms of performance. An important feature not treated here is concerning the management of the unknown disturbances while matching output constraints. The ideas presented in this paper could be as well used for extending usual methods (set range, funnel) to the case of non positive response systems.
6. REFERENCES Alamir, M. and G. Bornard (1995). Stability of a truncated infinite constrained receding horizon scheme: the general discrete nonlinear case. Automatica 31(9),1353-1356.
1086
Alamir, M. and N. Marchand (1999). Numerical stabilisation of nonlinear systems: exact theory and approximate numerical implementation. European Journal of Control 5(1), 8797. Bai, E-W. and S. Dasgupta (1996). A minimal k-step delay controller for robust tracking of non-minimum phase systems. Systems €1 Control Letters 28, 197-203. Bemporad, A., A. Casavola and E. Mosca (1997). Nonlinear control of constrained linear systems via predicitve reference management. IEEE Trans. on Automatic Control 42(32), 340-349. Bornard, G. and J.-P. Gauthier (1983). Commande multivariable en presence de contraintes de type inegalit. RAIRO, Autom. Syst. Anal. Control 17, 205-222. Commault, C. and J.-M. Dion (1982). structure at infinity of linear multivariable systems: a geometric approach. IEEE Trans. on Automatic Control 27, 693--696. Gomez, G. I. and G. C. Goodwin (1996). Integral constraints on sensitivity vectors for multivariable linear systems. Automatica 32(4),499-518. Horowitz, I. M., S. Oldak and O. Yaniv (1986). An important property of non-minimumphase multiple-input-multiple-output feedback systems. International Journal of Control 44(3), 677-688. Keerthi, S. S. and E. G. Gilbert (1988). Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations. Journal of Optimization Theory and Applications 57(2), 265-293. Prett, D. M. and C. E. Garcia (1988). Fundamental Process Control. Butterworths. Boston. Rawlings, J. B. and K. R. Muske (1993). Stability of constrained receding horizon control. IEEE Trans. on Automatic Control 38(10), 1512--6. Scokaert, P. O. M. and D. Q. Mayne (1998). Minmax model predictive control for constrained linear systems. IEEE Trans. on Automatic Control 43(8), 1136-1142. Wonham, W. M. (1985). Linear Multivariable Control - A Geometric Approach. SpringerVerlag. New York.