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Time-Domain Robust Stability Test Under Plant and Controller Interval Uncertainty
Copyright ® IFAC Robust Control Design. Prague. Czech Republic. 2000
TIME-DOMAIN ROBUST STABILITY TEST UNDER PLANT AND CONTROLLER INTERVAL UNCERTAINTY Pep Cuguer6· Sebastian Tornil· Teresa Escobet·· J ordi Saludes ••• Vicen~ Puig·
• Departament d'Enginyeria de Sistemes, Universitat Politecnica de Catalunya, Terrassa, Spain •• Departament d'Enginyeria de Sistemes, Universitat Politecnica de Catalunya, Manresa, Spain ... Departament de Matemdtica Aplicada 11, Universitat Politecnica de Catalunya, Terrassa, Spain
Abstract: Once a controller has been designed for a certain plant with any method available, one would like to know if that controller will perform well despite structured uncertainty in the plant. Furthermore, as controller implementation is concerned, one is also interested in its fragility (Keel and Bhattacharyya, 1997). In this paper, interval parametric uncertainty is considered. As an alternative to testing robust stability by polynomial methods (Bhattacharyya et al., 1995; Ackermann, 1993), one can perform a robust simulation of the time domain response of the closed loop taking into account plant and controller uncertainties. It is shown that, based on this simulation, it is possible to test the closed loop robust stability. Copyright ©2000 IFAC Keywords: robustness, fragility, uncertainty, intervals, envelopes, time-domain analysis
envelope is overbounded if it does include all the points belonging to every possible trajectory of all the systems in the model set, but not every point in it belongs to a trajectory of at least one system in the model set. The most interesting envelope is the exact one, which is neither under bounded nor bounded; it contains the points of every possible trajectory and every point it contains belongs to a possible trajectory of a system in the model set. However, the exact envelope is difficult to compute. In Robust Model-based Fault Detection, envelopes are used to robustly characterize faulty plant situations as they provide adaptive alarm thresholds for the measured variables of the plant (Puig et al., 1997; Armengol et al., 1999). IT the measure of any variable lies outside its corresponding envelope, it is considered that there is a fault in the system.
1. INTRODUCTION
To test the robustness of a controller design, the problem of simulating a closed loop in which plant and controller parameters are uncertain has to be addressed. The difficulty is that simulation results have to be obtained for an infinite set of models. To characterize the trajectories, given an input, of such uncertain models, the concept of envelope trajectory -or just envelope- of a variable can be used. An envelope of a variable at every time instant is an interval set that contains points of the time response of that variable for every model in the model set. An envelope is underbounded if every point inside it belongs to the trajectory of at least one system in the model set, but it does not include all the points of possible trajectories of all the systems in the model set. Conversely, an
207
To test robustness, every possible behaviour contained in the closed loop model set has to be considered, so it is of interest to obtain the least overbounded envelope and, if possible, the exact one.
such that if L ;;:: L min , then envelopes [xi", xtJ are stable. A window with length Lmin is called minimum stable window. It is also defined a minimum quasistationary window with length is approximately equal to the system settling time. Such a window only increases the initial uncertainty in less than 5% of the initial uncertainty.
2. ENVELOPE GENERATION In this paper, a linear discrete time-invariant system represented in state space form Xk+l
= Axk + BUk
Iterative and non-iterative envelope generation are nonlinear and non convex optimization problems which have to be solved with numerically reliable algorithms to ensure consistency. In (Puig et al., 1997; Puig et al., 1999) a global optimization based on interval arithmetic (Hansen, 1992) combined with bronch and bound has been used.
(1)
Yk =CXk +DUk
with its parameters bounded in intervals, A E [A-,A+) ,B E [B-,B+) will be considered. As explained in (Puig et al., 1997; Puig et al., 1999), the exact envelope for Xk at sample k, i.e. its interval bound [xi", x t], starting in an uncertain initial condition Xo E [xo, xt] can be generated by solving the optimization problem xi"
= min xl.:
To simplify notation let envelope dynamics (eq. 3) be expressed more compactly by the following interval equation
(2)
= min(Akxo + A k- 1 Buo xt
3. TESTING STABILITY BY ITERATIVE ENVELOPE GENERATION
+ ... + BUk-l)
= maxxk = max(Akxo
(4)
+ A k- 1 Buo + ... + BUk-d
where Ek denotes the envelope [xi", xtJ at time in xi" and xt, with bounds in A, B and Xo as restrictions.
k. To derive robust stability conclusions from a iteratively generated envelope it is necessary to have the complementary of the previously stated proposition.
This approach needs an increasing with time amount of computation to generate the envelopes, which makes it unsuitable for real time operation. As pointed in (Puig et al., 1997; Puig et al., 1999) this problem is due to the fact that the computation is always referred to the initial state Xo. The solution given there to overcome this problem was to generate the envelope iteratively by optimizing over a finite-time sliding window of length L which simplifies the previous optimization problem to
Proposition 2. Let eQuation 4 represent the envelope dynamics of an uncertain system. Let
(5)
(3)
xi" = min Xk
+ A L - 1 BUk-L + ... + BUk-l)
for a certain k. and a certain window length L. Suppose that Ek.and Ek.-L are non empty intervals and that input U is a step that remains constant for k > k. - L. Then the system is stable.
where the state uncertainty at the beginning of the window, Xk-Lhas taken the place ofthe k = 0 state uncertainty Xo at k = O.
PROOF. If we suppose that inputs remain constant forever we can describe envelope dynamics in time steps of L samples and denote it by
This solution relies on a result proved in (Saludes et al., 1997):
(6)
= min(ALxk_L
+
A L - 1 BUk-L
+ ... + BUk - d
xt = maxxk = max(ALxk_L
and apply it to equations 5 obtaining
Proposition 1. Let equation 3 define envelope dynamics. Then asymptotic stability of A implies the existence of a value Lmin for the window length L
(7)
208
1 .•
which, due to inclusion, are still true for q repetitive applications of W L
,----.----,----r---,----,-----,----r---,----,---,
> 0 •
(8) D.
The previous equation can be written DD~~-~,D~~,.-~~~~~-~~~-~~~~~~ .. -~~
(9)
TIIne(..:cnd)
Fig. 1. Window length 10, Controller uncertainty 0%
E k . - l +qL ~ E k . - l
q>O
" ,----,-----,----r-----;,----,-----,---.,...----,---,----,
Therefore, envelopes will remain bounded forever, which means, by their nature, that every system from the model set will have its trajectory bounded for a step input, so it will be stable. .,
The previous result can be summarized as follows: D.
Proposition 3. A linear time invariant uncertain system is stable if there exists a value L such that if two consecutive windows of length L are taken covering 2L samples of the step response of a system, then every envelope belonging to the second window does not exceed its corresponding envelope of the first one.
DD~~-~ID~~,.-~~~~~-~~~-3~.-~.~D-~ .. -~~ TIIM(-=cnd)
Fig. 2. Window length 10, Controller uncertainty 20% All the closed loop poles are placed at zp = ~, which yields for the nominal parameters of the controller rl = ~6' ro = /6·
Additionally, the existence of a minimum stable window length Lmin for a system, gives a measure of the extension of the time interval within which the parameters must remain invariant. This suggests how the envelopes can show the amount of tolerated time variance in the plant parameters within their corresponding intervals. Therefore, it seems that the system time-invariance condition could be relaxed.
4.1 Simulation results In the following simulations a fixed interval uncertainty of 20% around the nominal values of plant parameters has been considered. For controller parameters the same kind of uncertainty has been considered for 3 different values: 0%, 20% and 50%. Different windows length have also been considered. The simulation plots are ordered in the following fashion. The first 3 correspond to a sliding window length of 10 and values of 0%, 20% and 50% for the uncertainty of the controller parameters. The following 3 correspond to a sliding window length of 5. The last 3 ones correspond to a sliding window length of 3 and the same range of controller uncertainty.
4. EXAMPLE OF APPLICATION The above concepts will be illustrated by robustly simulating a simple closed loop of a discrete first order plant controlled by a PI regulator designed by pole placement. The transfer function of the process is _ _b_l_ Gp Z -
with nominal parameters b1 one of the PI regulator is r
(10)
al
= l~' al = ~ and the
G = rOz+rl
z-l
I
I:
As can be seen, reducing window length enlarges envelopes allowing less uncertainty in the controller, or makes them unstable -as it happens for every window of length 3-. It seems that the more
(11)
209
1.',------,-----,--,---,--,---,---,---.---,-----,
. 5,--,---.---,--,--,---,------,---.---,------,
.5
.,
,.' .,,
~~~-~ ,.--,~'-~~--~~-~~~~--~~~.,-~~
~.~~-~,.--,~'-~~-~~--~~~~-~~--.~,~~ T.... (1eCIlf1d)
Tm.(-xwd)
Fig. 3. Window length 10, Controller uncertainty
Fig. 6. Window length 5, Controller uncertainty
50%
50%
.,
••
~~~-~, . -~,~'-~~r-~~-~~ ~~ ~r-~-~.~,~~
••~~--,~.-~,~'-~~-~~~~~~~~--.~.-~.,~-!~
T ..... (seoond)
1.... (MCOnd)
Fig. 7. Window length 3, Controller uncertainty
Fig. 4. Window length 5, Controller uncertainty
0%
0%
1.5,------,-----,--,---,--,---,---,---,.---,-----, I.5,------,---,---.,---,---.--------,r---,---,---,----,
.,
Fig. 8. Window length 3, Controller uncertainty
Fig. 5. Window length 5, Controller uncertainty
20%
20%
5. CONCLUSIONS
uncertainty in the controller is allowed, the longer the window length has to be to have envelope stability (assuming that the closed-loop system remains stable). This can be interpreted as a decrease of tolerated time variance in the plant parameters.
A time domain technique for testing robust stability has been presented and illustrated with a simple example. This technique allows an iterative robust controller design process based on applying traditional, not necessarily robust, design tech-
210
ing window and global optimization. In: Proceedings of the European Control Conference ECC'99. Karlsruhe, Germany. Saludes, J., V. Puig and J. Quevedo (1997). Determination of window length for a new algorithm in adaptive threshold generation. In: TEMPUS Workshop SJEP 07759-94. Budapest, Hungary. 05
o ' o
10
1~
2'0
~
30
35
"'0
"5
50
TII'MI(~
Fig. 9. Window length 3, Controller uncertainty 50% niques whose results can be tested by simulation for further refinement.
6. ACKNOWLEDGEMENTS This paper is partially supported by the spanish CICYT under contracts TAP98-0585-C03-01 and TAP99-0748, and by the CIRIT of the Generalitat de Catalunya (ref. 1999SGRDOI34). The authors wish also to thank the support received by LEASICA (European Associated Laboratory on Intelligent Systems & Advanced Control).
7. REFERENCES Ackermann, J. (1993). Robust Control. Systems with Uncertain Physical Parameters. Springer-Verlag. Berlin, Germany. Armengol, Q., J. Vehf, L. Trave-Massuyes and M. A. Sainz (1999). Generation of errorbounded envelopes by means of modal interval analysis. In: Workshop on Applications of Interval Analysis to Systems and Control MISC'99. Girona, Spain. pp. 251-261. Bhattacharyya, S. P., H. Chapellat and L. H. Keel (1995). Robust Control. The Parametric Approach. Prentice Hall. New Jersey, USA. Hansen, E. (1992). Global Optimization Using Interval Analysis. Marcel Dekker. New York, USA. Keel, L. H. and S. P. Bhattacharyya (1997). R0bust, fragile, or optimal? IEEE 1ransactions on Automatic Control 42(8), 1098-1105. Puig, V., J. Saludes and J. Quevedo (1997). Applications in fault detection and diagnosis of a new algorithm for adaptive threshold generation. In: TEMPUS Workshop SJEP 0775994. Budapest, Hungary. Puig, V., J. Saludes and J. Quevedo (1999). A new algorithm for adaptive threshold generation in robust fault detection based on a slid-
211
TIME-DOMAIN ROBUST STABILITY TEST UNDER PLANT AND CONTROLLER INTERVAL UNCERTAINTY Pep Cuguer6· Sebastian Tornil· Teresa Escobet·· J ordi Saludes ••• Vicen~ Puig·
• Departament d'Enginyeria de Sistemes, Universitat Politecnica de Catalunya, Terrassa, Spain •• Departament d'Enginyeria de Sistemes, Universitat Politecnica de Catalunya, Manresa, Spain ... Departament de Matemdtica Aplicada 11, Universitat Politecnica de Catalunya, Terrassa, Spain
Abstract: Once a controller has been designed for a certain plant with any method available, one would like to know if that controller will perform well despite structured uncertainty in the plant. Furthermore, as controller implementation is concerned, one is also interested in its fragility (Keel and Bhattacharyya, 1997). In this paper, interval parametric uncertainty is considered. As an alternative to testing robust stability by polynomial methods (Bhattacharyya et al., 1995; Ackermann, 1993), one can perform a robust simulation of the time domain response of the closed loop taking into account plant and controller uncertainties. It is shown that, based on this simulation, it is possible to test the closed loop robust stability. Copyright ©2000 IFAC Keywords: robustness, fragility, uncertainty, intervals, envelopes, time-domain analysis
envelope is overbounded if it does include all the points belonging to every possible trajectory of all the systems in the model set, but not every point in it belongs to a trajectory of at least one system in the model set. The most interesting envelope is the exact one, which is neither under bounded nor bounded; it contains the points of every possible trajectory and every point it contains belongs to a possible trajectory of a system in the model set. However, the exact envelope is difficult to compute. In Robust Model-based Fault Detection, envelopes are used to robustly characterize faulty plant situations as they provide adaptive alarm thresholds for the measured variables of the plant (Puig et al., 1997; Armengol et al., 1999). IT the measure of any variable lies outside its corresponding envelope, it is considered that there is a fault in the system.
1. INTRODUCTION
To test the robustness of a controller design, the problem of simulating a closed loop in which plant and controller parameters are uncertain has to be addressed. The difficulty is that simulation results have to be obtained for an infinite set of models. To characterize the trajectories, given an input, of such uncertain models, the concept of envelope trajectory -or just envelope- of a variable can be used. An envelope of a variable at every time instant is an interval set that contains points of the time response of that variable for every model in the model set. An envelope is underbounded if every point inside it belongs to the trajectory of at least one system in the model set, but it does not include all the points of possible trajectories of all the systems in the model set. Conversely, an
207
To test robustness, every possible behaviour contained in the closed loop model set has to be considered, so it is of interest to obtain the least overbounded envelope and, if possible, the exact one.
such that if L ;;:: L min , then envelopes [xi", xtJ are stable. A window with length Lmin is called minimum stable window. It is also defined a minimum quasistationary window with length is approximately equal to the system settling time. Such a window only increases the initial uncertainty in less than 5% of the initial uncertainty.
2. ENVELOPE GENERATION In this paper, a linear discrete time-invariant system represented in state space form Xk+l
= Axk + BUk
Iterative and non-iterative envelope generation are nonlinear and non convex optimization problems which have to be solved with numerically reliable algorithms to ensure consistency. In (Puig et al., 1997; Puig et al., 1999) a global optimization based on interval arithmetic (Hansen, 1992) combined with bronch and bound has been used.
(1)
Yk =CXk +DUk
with its parameters bounded in intervals, A E [A-,A+) ,B E [B-,B+) will be considered. As explained in (Puig et al., 1997; Puig et al., 1999), the exact envelope for Xk at sample k, i.e. its interval bound [xi", x t], starting in an uncertain initial condition Xo E [xo, xt] can be generated by solving the optimization problem xi"
= min xl.:
To simplify notation let envelope dynamics (eq. 3) be expressed more compactly by the following interval equation
(2)
= min(Akxo + A k- 1 Buo xt
3. TESTING STABILITY BY ITERATIVE ENVELOPE GENERATION
+ ... + BUk-l)
= maxxk = max(Akxo
(4)
+ A k- 1 Buo + ... + BUk-d
where Ek denotes the envelope [xi", xtJ at time in xi" and xt, with bounds in A, B and Xo as restrictions.
k. To derive robust stability conclusions from a iteratively generated envelope it is necessary to have the complementary of the previously stated proposition.
This approach needs an increasing with time amount of computation to generate the envelopes, which makes it unsuitable for real time operation. As pointed in (Puig et al., 1997; Puig et al., 1999) this problem is due to the fact that the computation is always referred to the initial state Xo. The solution given there to overcome this problem was to generate the envelope iteratively by optimizing over a finite-time sliding window of length L which simplifies the previous optimization problem to
Proposition 2. Let eQuation 4 represent the envelope dynamics of an uncertain system. Let
(5)
(3)
xi" = min Xk
+ A L - 1 BUk-L + ... + BUk-l)
for a certain k. and a certain window length L. Suppose that Ek.and Ek.-L are non empty intervals and that input U is a step that remains constant for k > k. - L. Then the system is stable.
where the state uncertainty at the beginning of the window, Xk-Lhas taken the place ofthe k = 0 state uncertainty Xo at k = O.
PROOF. If we suppose that inputs remain constant forever we can describe envelope dynamics in time steps of L samples and denote it by
This solution relies on a result proved in (Saludes et al., 1997):
(6)
= min(ALxk_L
+
A L - 1 BUk-L
+ ... + BUk - d
xt = maxxk = max(ALxk_L
and apply it to equations 5 obtaining
Proposition 1. Let equation 3 define envelope dynamics. Then asymptotic stability of A implies the existence of a value Lmin for the window length L
(7)
208
1 .•
which, due to inclusion, are still true for q repetitive applications of W L
,----.----,----r---,----,-----,----r---,----,---,
> 0 •
(8) D.
The previous equation can be written DD~~-~,D~~,.-~~~~~-~~~-~~~~~~ .. -~~
(9)
TIIne(..:cnd)
Fig. 1. Window length 10, Controller uncertainty 0%
E k . - l +qL ~ E k . - l
q>O
" ,----,-----,----r-----;,----,-----,---.,...----,---,----,
Therefore, envelopes will remain bounded forever, which means, by their nature, that every system from the model set will have its trajectory bounded for a step input, so it will be stable. .,
The previous result can be summarized as follows: D.
Proposition 3. A linear time invariant uncertain system is stable if there exists a value L such that if two consecutive windows of length L are taken covering 2L samples of the step response of a system, then every envelope belonging to the second window does not exceed its corresponding envelope of the first one.
DD~~-~ID~~,.-~~~~~-~~~-3~.-~.~D-~ .. -~~ TIIM(-=cnd)
Fig. 2. Window length 10, Controller uncertainty 20% All the closed loop poles are placed at zp = ~, which yields for the nominal parameters of the controller rl = ~6' ro = /6·
Additionally, the existence of a minimum stable window length Lmin for a system, gives a measure of the extension of the time interval within which the parameters must remain invariant. This suggests how the envelopes can show the amount of tolerated time variance in the plant parameters within their corresponding intervals. Therefore, it seems that the system time-invariance condition could be relaxed.
4.1 Simulation results In the following simulations a fixed interval uncertainty of 20% around the nominal values of plant parameters has been considered. For controller parameters the same kind of uncertainty has been considered for 3 different values: 0%, 20% and 50%. Different windows length have also been considered. The simulation plots are ordered in the following fashion. The first 3 correspond to a sliding window length of 10 and values of 0%, 20% and 50% for the uncertainty of the controller parameters. The following 3 correspond to a sliding window length of 5. The last 3 ones correspond to a sliding window length of 3 and the same range of controller uncertainty.
4. EXAMPLE OF APPLICATION The above concepts will be illustrated by robustly simulating a simple closed loop of a discrete first order plant controlled by a PI regulator designed by pole placement. The transfer function of the process is _ _b_l_ Gp Z -
with nominal parameters b1 one of the PI regulator is r
(10)
al
= l~' al = ~ and the
G = rOz+rl
z-l
I
I:
As can be seen, reducing window length enlarges envelopes allowing less uncertainty in the controller, or makes them unstable -as it happens for every window of length 3-. It seems that the more
(11)
209
1.',------,-----,--,---,--,---,---,---.---,-----,
. 5,--,---.---,--,--,---,------,---.---,------,
.5
.,
,.' .,,
~~~-~ ,.--,~'-~~--~~-~~~~--~~~.,-~~
~.~~-~,.--,~'-~~-~~--~~~~-~~--.~,~~ T.... (1eCIlf1d)
Tm.(-xwd)
Fig. 3. Window length 10, Controller uncertainty
Fig. 6. Window length 5, Controller uncertainty
50%
50%
.,
••
~~~-~, . -~,~'-~~r-~~-~~ ~~ ~r-~-~.~,~~
••~~--,~.-~,~'-~~-~~~~~~~~--.~.-~.,~-!~
T ..... (seoond)
1.... (MCOnd)
Fig. 7. Window length 3, Controller uncertainty
Fig. 4. Window length 5, Controller uncertainty
0%
0%
1.5,------,-----,--,---,--,---,---,---,.---,-----, I.5,------,---,---.,---,---.--------,r---,---,---,----,
.,
Fig. 8. Window length 3, Controller uncertainty
Fig. 5. Window length 5, Controller uncertainty
20%
20%
5. CONCLUSIONS
uncertainty in the controller is allowed, the longer the window length has to be to have envelope stability (assuming that the closed-loop system remains stable). This can be interpreted as a decrease of tolerated time variance in the plant parameters.
A time domain technique for testing robust stability has been presented and illustrated with a simple example. This technique allows an iterative robust controller design process based on applying traditional, not necessarily robust, design tech-
210
ing window and global optimization. In: Proceedings of the European Control Conference ECC'99. Karlsruhe, Germany. Saludes, J., V. Puig and J. Quevedo (1997). Determination of window length for a new algorithm in adaptive threshold generation. In: TEMPUS Workshop SJEP 07759-94. Budapest, Hungary. 05
o ' o
10
1~
2'0
~
30
35
"'0
"5
50
TII'MI(~
Fig. 9. Window length 3, Controller uncertainty 50% niques whose results can be tested by simulation for further refinement.
6. ACKNOWLEDGEMENTS This paper is partially supported by the spanish CICYT under contracts TAP98-0585-C03-01 and TAP99-0748, and by the CIRIT of the Generalitat de Catalunya (ref. 1999SGRDOI34). The authors wish also to thank the support received by LEASICA (European Associated Laboratory on Intelligent Systems & Advanced Control).
7. REFERENCES Ackermann, J. (1993). Robust Control. Systems with Uncertain Physical Parameters. Springer-Verlag. Berlin, Germany. Armengol, Q., J. Vehf, L. Trave-Massuyes and M. A. Sainz (1999). Generation of errorbounded envelopes by means of modal interval analysis. In: Workshop on Applications of Interval Analysis to Systems and Control MISC'99. Girona, Spain. pp. 251-261. Bhattacharyya, S. P., H. Chapellat and L. H. Keel (1995). Robust Control. The Parametric Approach. Prentice Hall. New Jersey, USA. Hansen, E. (1992). Global Optimization Using Interval Analysis. Marcel Dekker. New York, USA. Keel, L. H. and S. P. Bhattacharyya (1997). R0bust, fragile, or optimal? IEEE 1ransactions on Automatic Control 42(8), 1098-1105. Puig, V., J. Saludes and J. Quevedo (1997). Applications in fault detection and diagnosis of a new algorithm for adaptive threshold generation. In: TEMPUS Workshop SJEP 0775994. Budapest, Hungary. Puig, V., J. Saludes and J. Quevedo (1999). A new algorithm for adaptive threshold generation in robust fault detection based on a slid-
211