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Robust Control for Linear Systems with Actuator Rate and Amplitude Saturation Using QFT
ELSEVIER
Copyright © IFAC Robust Control Design Milan, Italy, 2003
IFAC
ROBUST CONTROL FOR LINEAR SYSTEMS WITH ACTUATOR RATE AND AMPLITUDE SATURATION USING QFT Jose Carlos Moreno, • Alfonso Baiios, •• and Manuel Berenguel'
• Dpto. Lenguajes y Computacion, Escuela Politecnica Superior Universidad de Almeria, 04120 Almeria, Spain •• Dpto. Informatica y Sistemas, Facultad de Informatica Universidad de Murcia, 30071 Murcia, Spain
Abstract: In this paper a new design methodology to cope with global stability in presence of rate and amplitude saturation in systems with uncertain plants is proposed. The design method proposed provides an accurate graphic design procedure in frequency domain. To deal with the problem a 4DoF control scheme, two (F and G) designed in a first step with QFT so that the uncertainty is taken into account for the linear operation mode, and the other two (Ha and H v ) designed in a second step to deal with the non-linear operation mode. Copyright © 2003 IFAC
Keywords: Robust control, control system synthesis, bounded control, global stability.
suriya, 2001)), and when the control signal is saturated in amplitude and after in rate ((Kapasouris and Athans, 1990),(Tyan and Bernstein, 1997)). In this paper the first configuration is adopted, but results may be extrapolated to the other configuration. Using a 4DoF scheme as in figure 1, where the rate saturation is expressed as a modified amplitude saturation, conditions over Ha and H v are derived which will assure the global stability of the system, and a method to design these compensators is proposed, supposing that F and G compensators have been previously designed as if there not exist saturation elements in the system, in a previous stage in which QFT has been used in order to take the plant uncertainty into account for the linear operation mode of system. For simplicity, a direct differentiator s is used to model the rate constraint, as in (Horowitz, 1984). Otherwise, a dynamic feedback loop may be used to model the rate constraint,
1. INTRODUCTION
In this paper the global stability problem for QFT designs for linear systems with rate and amplitude actuator saturation is analyzed. This problem is also discussed in (Hui, 2001), where a 3DoF scheme for the case in which there not exist plant uncertainty is proposed. However, the frequency domain design method proposed in (Hui, 2001) does not guarantee the global stability of system because a relaxed form of a sufficient condition (Freudenberg and Looze, 1988) very conservative to ensure global stability is used. In (Wu and Jayasuriya, 2001) the problem is discussed for the uncertain case, however global stability aspects are not considered. In relation to the placement of the amplitude saturation with respect to the rate saturation, two cases may be considered: when the control signal is saturated in rate before in amplitude (case considered in (Horowitz, 1984),(Liao and Horowitz, 1986), (Hui, 2001), (Wu and Jaya91
templates. In particular, X°(jw) = p°(jw)G(jw) represents the nominal point in the ni-template for frequency w. The ni-template and the I-template sets will allow us to deal with the uncertainty in an explicit form in the process design for the shaping of anti-windup compensator {Ha, H v }, in the second step of classical anti-windup paradigm, in which the non-linear operation mode is taken into account. Note that the shape of the nI-template is the same as the corresponding I-template, only the placement in NP is different. So that, computational effort is not required to computation of £fnl(W) if £f1(W) has been previously computed to design F and G compensators with QFT.
Fig. 1. A 4DoF control scheme with amplitude and rate actuator saturation. as in (Tyan and Bernstein, 1997). The paper is organized as follows. In section 2 some important definitions are introduced. In section 3 a set of results is proposed to assure the global stability of system in figure 1. In section 4 a design method is proposed to cope with the plant uncertainty. In section 5 an illustrative example is sho~n, and finally some conclusions are outlined in section 6. All proofs of the proposed results can be found in (Moreno, 2003).
Definition 3. For each frequency w, a boundary
b(w) is defined with respect to four vertical lines Plo (w), Pro (w), PI" (W) and Pr" (w), as the boundary with a region in NP so that if X°(jw) lies within
2. PRELIMINARIES
that zone, then one condition is satisfied (condition depending the context in which the boundary is used) for all plants P in p.
In figure 1 the linear block P represents a SISO LT! stable plant with transfer function P( s) belonging to a set of plants p. The compensator consists of four blocks: F(s), the precompensator; G(s), the feedback compensator, and {Ha(s), Hv(s)}, the anti-windup compensator. Assuming ideal saturation, the saturating elements outputs, Sat a and Sat v in figure 1, are given by the function
y(t) =
x(t) if x(t) E [min, max] min if x(t) < min { max if x(t) > max
Definition 4. For each frequency w functions br(w) and bl(w) are defined as
(1) so that
where x(t) = xa(t) and y(t) = Ya(t) for the Sat a element, and where x(t) = xv(t) and y(t) = yv(t) for the Sat v element, with xa(t) and xv(t) the Sat a and the Sat v inputs respectively, and with Ya(t) and yv(t) the Sat a and the Satv outputs. Amplitude saturation limits min = Y1nin and max = Y1nax, and rate saturation limits min = V 1n in and max = V 1nax are assumed. In the following, some definitions to be used afterwards are introduced.
(4) The equation (2) defines an allowed zone between
Plo(w) and Pr" (w) degrees, and (3) defines another allowed region between PI" (w) and Pro (w) degrees. Figure 4(a) shows an example of this type of boundary.
Definition 1. A ni-template (template for nonlinear mode) for each frequency w, £fnl(w), is defined as a set of complex value given by £fnl(w) ~
{P(jw)G(jw) : P E
3. STABILITY ANALYSIS FOR COMPENSATED SYSTEM
d.
In order to analyze the system stability is more convenient to transform the system in figure 1 to an equivalent system (from the stability point of view), consisting in a feedback interconnection of a linear block, including all linear dynamics present in the loop, and a non-linear block representing the saturating elements Sat a and Sat v . Considering signals Yv, Ya and r in figure 1 as system inputs, in order to compute Xv and X a as a function of them, and using ii to note the Laplace transform for a time signal a,
Definition 2. A I-template (template for linear mode) for each frequency w, £f1(W), is defined as a set of complex values given by £fl(w) ~ {P(jw) :
PEp}. Both of them are considered in the Nichols plane (NP), where complex numbers are represented as magnitude in decibels vs phase in degrees (generally in the [-360°,0°] range). For a nominal pO E p, a nominal point is obtained for both types of
92
Proposition 6, If conditions (i)-(iv) in Proposition 5 and the following conditions are satisfied
"
~ .
A..
all".
,
(v) U(G)i > U(P)i + 1, and (vi) U(G)i + 1 = U(Hv) and U(G)i + 1 = "(Ha), U(G)i + 1 = U(Hv) Y U(G)i = U(Ha), U(G)i + 1 = U(Ha) y U(G)i = U(Hv).
"
Fig. 2. Equivalent system with matrix blocks.
[ ~v] [1:~~V 1-:~~V] [~v] + =
Xa
~Ya s(1 + Ha) 1 + Ha
then the stability of K and the stability of Rare assured, but if the system in figure 1 is stable then an offset will appear in the output.
1
+[1
(5)
;~" i ~ K [t 1+ Ri ]
In the following, a version of the result based in robust stability with multipliers developed in (Moreno,2003) (using a previous result in (Zames and Falb, 1968» for SISO systems, to deal with amplitude saturation, is proposed to deal with simultaneous amplitude and rate saturation.
In figure 2 the resulting scheme is shown, where K and R blocks are given by equation (5) and with SATS non-linear block given by
SATS = [sat v
o
0 ] Sat a
(6)
Theorem 7. For the scheme of figure 2, if
A necessary condition for the system stability is that R in equation (5) must be stable. The stability for a transfer functions matrix given in (Skogestad and Postlethwaite, 1996) is used. The notation tt(')i is introduced to note the number of integrators in a transfer function. Defining functions Fv and Fa given by Fv="'--~
1 1 +sHv
(7)
F =_1_
(8)
a
I+H a
(i) The non-linearity SATS is diagonal, odd, monotone nondecreasing and belonging to slope
[O,lj, (ii) The linear part K(s) is stable, (iii) There exists some multiplier Z(s) = diag( Zl(S),Z2(S» (possibly non causal) and X = diag(X 1 , X 2) > 0, so that (iii-a) The impulse response Zi(t) for Zi(S) satisfies that IIzi(t)lll < Xi, Le
1:
00
and using equation (5) for matrix K, then K
=
[1 -
F v -SPGFv] Fa/s 1- Fa
IZi(t) Idt < Xi, for i
= 1,2
(iii-b) In frequency domain, for all frequency wand for some { > 0,
(9)
(X - Z(jw»(K(jw) + 1)+ + (K*(jw) + l)(X - Z*(jw»
is obtained. The matrix K stability, including the linear dynamics of system in figure 1, is key to apply different frequency domain stability criteria. The next Proposition sets necessary conditions to assure the stability for Kin (9), using the previous stability definition.
~
d
(10)
(iv) If SATS is even, then in addition it must be satisfied that
Zi(t)
~
0, Vt E (-00,00) with i
= 1,2.
then the interconnection is stable.
Proposition 5. If the following is satisfied
In the following Proposition the Theorem 7 is adapted to the case handled in this paper.
(i) U(P)i:S 1 VP E p, (ii) The number of zeros in origin for function Fa, given by (8), is greater or equal than one, (iii) All of poles of function Fa in (8) belong to open LHP (Left Half Plane excluding the origin), (iv) All of poles of function F v in (7) belong to open LHP, (v) U(G)i = U(P)i + 1 VP E p, (vi) U(G)i = U(Hv)i, and (vii) U(G)i = tt(Ha};.
Proposition 8, If there exist functions Fv and Fa (given by (7) and (8) respectively), and multipliers Zl and Z2, so that the next conditions are satisfied
(i) If Zl(t) and Z2(t) are the impulse response of Zl (s) and Z2 (s) respectively +00 ;
-00
;
-00
!zl(t)ldt < 1
+00
then the stability of K in (9) and the stability of R in (5) is assured, and if the system in figure 2 is stable then no offset will appear in the system output.
I Z2(t)!dt
<1
(ii) Conditions in Proposition 5 or conditions in Proposition 6 are satisfied 93
(Hi) -Fv(jw) "I- -2 + ja/(l - Zl(jW») "la E R,Vw > 0 (iv) -Fa(jw) "I- -2 + ja/(l - Z2(jW») "la E IR,Vw > 0 (v) Re( -P(jw)G(jw)Fv (jw)jw)Re(2 - Zl (jw) Z2(jW» 2: Im(-P(jw)G(jw)Fv (jw)jw)Im(2Zl(jW) - Z2(jW» Vw > 0 VP E P (vi) Re(Fa (jw)/(jw»Re(2 - Zl (jw) - Z2(jW» 2: Im(Fa(jw)/(jw»Im(2-Z1(jW)-Z2(jw» Vw >
.....1
Fig. 3. Boundaries over L°(jw) for frequency w in order to guarantee absolute stability for system without saturating elements.
o (vii) If SATS is even, then in addition the following condition must be satisfied
Using Z(s) = 0 a version of multivariable Circle Criterion is obtained, which is enunciated in the next Corollary.
Zl(t),Z2(t) 2: 0, "It E (-00,00) then the system in figure 2, with K given by (9), R given by (5) and SATS given by (6), is stable.
Corollary 10. If there exist functions Fa and Fv (given by (8) and (7) respectively) so that the following conditions are satisfied
4. CONTROL SYSTEM DESIGN
(i) Conditions in Proposition 5 or conditions in In this section a design method is proposed, taking the uncertainty of the plant into account, to choose compensators Ha and H v in figure 1, supposing that F and G have been previously designed using QFT, so that the uncertainty has been taken into account in the design process but only for the linear operation mode. In the following Corollary a result easier to check than in Proposition 8 is established.
Proposition 6 are satisfied, (ii) Re(Fv(jw» < 2 Vw > 0 (Hi) Re(Fa(jw» < 2 Vw > 0 (iv) ~P,w,w- ,w+ with w > O,w- < w < w+ and PEp: Im(P(jw-)G(jw-)Fv(jw-» "Io and Im(P(jw+)G(jw+)Fv(jw+» "I- 0 and Im(P(jw)G(jw)Fv(jw» = 0 (v) ~,w- ,w+ with w > 0 and w- < w < w+ : Im(Fa(jw-» "I- 0 and Im(Fa(jw+» "Io and Im(Fa(jw» = 0
Corollary 9. If there exist functions Fa and Fv (given by (8) and (7) respectively), and a multiplier Z (s) so that the following conditions are satisfied
then the system in figure 2, with K given by (9), R given by (5) and SATS given by (6), is stable.
(i) If z(t) is the impulse response of Z(s) +00
1
-00
Iz(t)ldt < 1,
(ii) Conditions in Proposition 5 or conditions in Proposition 6 are satisfied, (Hi) -Fv(jw) "I- -2 + ja/(1 - Z(jw» "la E R,Vw > 0 (iv) -Fa(jw) "I- -2 + ja/(1 - Z(jw» "la E IR,Vw > 0 (v) ~a,P,w,w-,w+ with w > O,w- < w < w+,a E IR and PEp: P(jw-)G(jw-)Fv(jw-) "Ia/(l- Z(jw-» and P(jw+)G(jw+)Fv(jw+) "Ia/(I-Z(jw+» and P(jw)G(jw)Fv(jw) = a/(IZ(jw» (vi) ~a,w,w- ,w+ with w > O,w- < w < w+ and a E IR: Fa(jw-) "I- a/(I-Z(jw-» and Fa (jw+) "I- a/(1 - Z(jw+» and Fa(jw) = a/(1 - Z(jw» (vii) If SATS is even, then in addition the following must be satisfied
If H v = 0 is supposed (the rate saturation effect is not compensated) and Ha(s) = l/s, a restriction over G compensator is obtained (which must be incorporated to the set of restrictions used in the loop shaping stage of QFT for the linear operation mode). This is established in the following Corollary.
Corollary 11. If the following conditions are satisfied
(i) H v = 0, (ii) Ha(s) = l/s, (iii) U(P); = 0 VP E p, (iv) ~(G); = 1, and (v) The system, in linear operation mode (without considering saturating elements), is absolutely stable for all plant P in p,
then system in figure 2, with K given by (9), R given by (5) and SATS given by (6), is stable. The result in Corollary 11 will provide a set of constraints over the phase of function L°(jw) similar to shown in figure 3. So that simply this set of boundaries needs to be added to the set of
Z(t) 2: 0, "It E (-00,00) then the system in figure 2, with K given by (9), R given by (5) and SATS given by (6), is stable.
94
P(s) = 0.2 (s2 +26W1s+w?) (S2 +26W1s+w?) s2 + 26w2S + w~ s2 + 26w2S + w~ (12) and 6 E [0.5,O.6J. In the linear mode operation the system is controlled using a Proportional-Integral law (PI) given by with
(a) Circle Criterion
(b) Multipliers
= 0.2115, W2 = 0.0473, ~l E [0.2,OAJ
{F(s),C(s)}
Fig. 4. Boundary over X°(jw) for frequency w in order to assure the global stability of system.
= {I, 100S;10
4 }
(13)
and the nominal plant po is given by 6 = 0.2 and b = 0.5. Amplitude saturation limits are assumed to be Ymax = 0.5 and Ymin = -0.5, and rate saturation limits are given by V max = 0.1 and Vmin = -0.1. In this example the set of designing frequencies W = {O.OI, 0.04, 0.2, 0.35} rad/s is used. Using the control law in equation (13), Ha(s) = l/s and Hv(s) = 0, it may be probed that the system is absolutely stable as a direct application of result in Corollary 11. Conditions (i)-(iv) in Corollary 11 are satisfied. In figure 5(a) the Nichols diagram of function X°(jw) = L°(jw) (with D = 1) is shown, as well as ni-templates and boundaries bl(w) for each frequency w in W. As it may be observed, function X°(jw) satisfies all restrictions. The global stability of the system is assured for all values of uncertain parameters 6 and 6 in (12). If in place of using the control law in (13) for the linear mode operation, the following is used (as it is proposed in (Doyle et al., 1987»
boundaries derived from specifications for linear operation mode, in the loop shaping stage in QFT, which will assure the closed loop system stability for all plants P in p, in presence of rate and amplitude saturation elements. Note that the Corollary 9 states less conservative conditions than Corollaries 10 and 11, because an additional degree of freedom is allowed, the multiplier Z (s), to assure the system stability. Condition (iii) in Corollary 9 allows us to rotate the not allowed locus for Fv function in condition (ii) in Corollary 10, a value given by 1 - Z(jw) for each frequency w. The same occurs with conditions (iv) and (vi) in Corollary 9 and conditions (iii) and (v) in Corollary 10 for the case of Fa, and with condition (v) in Corollary 9 and condition (iv) in Corollary 10. Let the function X given by
X(s) =P(s)C(s)Fv(s) = =P(s)C(s)D(s) with PEp
Wl
(11)
{F(s),C(s)} =
with function Fv given by (7). It is supposed that {F, C} compensator was designed in a previous step. Note that shifting of ni-template for frequency win NP is equivalent to add ID(jw)1 decibels and Angle(D(jw» degrees to each point in ~1(W), or equivalently to the nominal X°(jw), assuming this as representative of the rest of points. So the introduction of poles and zeros in D in (11) is equivalent to shift the ni-templates. When D is designed the computation of H v is straightforward from (7). Then zones in NP must be defined so that if XO lies within them, then stability may be concluded. These zones can be defined using four vertical lines as shown in figure 4(a). For a more detailed example see section 5.
{I, 100Ss+ 1O}
(14)
and Ha(s) = l/s and Hv(s) = 0 are taken, then the function X°(jw) with D(s) = 1 does not fulfill the constraints, so that it is necessary to use an adequate compensator H v . A possible choice for Fv is Fv(s) = 2.5s + 1 (15) 4.975s + 1 which introduces an increase in the phase approximately of 20 degrees around w = 0.3 rad/s. Figure 5(b) shows boundaries bl(w) and ni-templates ~nl(W) for each frequency w in W, shifted by the shaping D = Fv , and function X°(jw) which, as it may be observed, fulfills all constraints. So, as Fv satisfies conditions (i) and (ii) in Corollary 10, the global stability of system is assured for all values of parameters 6 and 6 in (12).
Using the result in Corollary 9 the set of four vertical lines may be shifted in the NP using a multiplier Z (s). In figure 4(b) a typical boundary b(w) using this result is shown. For a more detailed example see section 5.
Finally, the global stability for design given by the control law for linear operation mode in (14) and Hv(s) = 0 (supposing that Ha(s) = l/s) may be probed too. Choosing the multiplier Z (s) = 1 Fv(s) with Fv given by (15), a modification in the location of boundaries is introduced, such that the system for F v = 1 (Hv = 0) satisfies all constraints. Conditions (i)-(iv) and (v) in Corollary 9 are satisfied. Restrictions on phase of X°(jw) are fulfilled too, as shown in figure 5(c). So, the global stability of system may be concluded. Note that
5. AN ILLUSTRATIVE EXAMPLE The plant considered for this example has been previously used in (Doyle et al., 1987), but uncertainty has been introduced in two parameters in order to study the proposed method. 95
grants QUI99-0663-C02-02, DPI2000-1218-C0403, DPI2001-2380-C02-02, and DPI2002-04375C03-03.
..1:
J:
:j!
;
"
/--
-*:
4:
'1.:
REFERENCES
0,
(a) D(s)
=1
i:
(b) D(s)
= Fv
Doyle, J. C., R. S. Smith and D. F. Enns (1987). Control of plants with input saturation nonlinearities. In: Proceedings of the 1987 American Control Conference. Vol. 145. Minneapolis. pp. 1034-1039. Freudenberg, J. S. and D. P. Looze (1988). Frequency Domain Properties of Scalar and Multivariable Feedback Systems. Springer Verlag. Berlin. Horowitz, I. M. (1984). Feedback systems with rate and amplitude limiting. International Journal of Control 40, 1215-1229. HUi, K. (2001). Design of rate and amplitude saturation compensators using three degrees of freedom controllers. International Journal of Control 74, 91-1Ol. Kapasouris, P. and M. Athans (1990). Control systems with rate and magnitude saturation for neutrally stable open loop systems. In: Proceedings of 29th IEEE Conference in Decision and Control. pp. 3404-3409. Liao, Y. K. and I. M. Horowitz (1986). Unstable uncertain plants with rate and amplitude saturations. International Journal of Control 44, 1147-1159. Moreno, J. C. (2003). Robust control of systems with input constraints. PhD thesis. Dept. of Computer Science and Systems. University of Murcia. Moreno, J. C., A. Baiios and M. Berenguel (2002). Robust control for linear systems with actuator saturation using qft. Submitted. Skogestad, S. and I. Postlethwaite (1996). Multivariable Feedback Control. Wiley. New York. Tyan, F. and D. S. Bernstein (1997). Dynamic output feedback compensation for linear systems with independent amplitude and rate saturations. International Journal of Control 67,89-116. Wu, W. and S. Jayasuriya (2001). A qft design methodology for feedback systems with input rate or amplitude and rate saturation. In: Proceedings of the 2001 American Control Conference. Arlington, VA. pp. 376-383. Zames, G. and P. L. Falb (1968). Stability conditions for systems with monotone and slope-restricted nonlinearities. SIAM Journal 6(1),89-108.
~
...•.• . :
~t
-':
,
(c) D(s) tiplier
= 1 with mul-
Fig. 5. ~nl(W) with w in Wand nominal X°(jw). in this example the value for 1- Z (s) is the same as previous Fv , so that we could think that the use of Fv or the multiplier is totally equivalent. However there are two important differences. In first place, the fact of probe the stability of system without using Fv implies that H v = 0, and in the second place, using Fv implies an advantage over the use of the multiplier Z (s) because there are no restrictions over the L 1 - norm of impulse response of Fv and there are restrictions over L 1 - norm of impulse response of Z. Furthermore, the use of the multiplier does not imply to exclude the use of Fv , both of them may be used together (as it is deduced, for example, in Corollary 10), providing the proposed method with great flexibility.
6. CONCLUSIONS In this paper the problem derived from the presence of rate and amplitude saturation in the control system with an uncertain plant is analyzed from a stability point of view. To deal with the problem a 4DoF control scheme, two (F and G) designed in a first step with QFT so that the uncertainty is taken into account for the linear operation mode, and the other two (Ha and Hv ) designed in a second step to deal with the non-linear operation mode. Taking as a basis the Zames-Falb result about stability with multipliers, a set of results and an algorithm to design Ha and Hv for an uncertain plant assuring the global stability of system in the presence of rate and amplitude saturation are proposed.
ACKNOWLEDGEMENTS The authors would like to thank the Spanish Comision Interministerial de Ciencia y Tecnologfa (CICYT) for partially funding this work under 96
Copyright © IFAC Robust Control Design Milan, Italy, 2003
IFAC
ROBUST CONTROL FOR LINEAR SYSTEMS WITH ACTUATOR RATE AND AMPLITUDE SATURATION USING QFT Jose Carlos Moreno, • Alfonso Baiios, •• and Manuel Berenguel'
• Dpto. Lenguajes y Computacion, Escuela Politecnica Superior Universidad de Almeria, 04120 Almeria, Spain •• Dpto. Informatica y Sistemas, Facultad de Informatica Universidad de Murcia, 30071 Murcia, Spain
Abstract: In this paper a new design methodology to cope with global stability in presence of rate and amplitude saturation in systems with uncertain plants is proposed. The design method proposed provides an accurate graphic design procedure in frequency domain. To deal with the problem a 4DoF control scheme, two (F and G) designed in a first step with QFT so that the uncertainty is taken into account for the linear operation mode, and the other two (Ha and H v ) designed in a second step to deal with the non-linear operation mode. Copyright © 2003 IFAC
Keywords: Robust control, control system synthesis, bounded control, global stability.
suriya, 2001)), and when the control signal is saturated in amplitude and after in rate ((Kapasouris and Athans, 1990),(Tyan and Bernstein, 1997)). In this paper the first configuration is adopted, but results may be extrapolated to the other configuration. Using a 4DoF scheme as in figure 1, where the rate saturation is expressed as a modified amplitude saturation, conditions over Ha and H v are derived which will assure the global stability of the system, and a method to design these compensators is proposed, supposing that F and G compensators have been previously designed as if there not exist saturation elements in the system, in a previous stage in which QFT has been used in order to take the plant uncertainty into account for the linear operation mode of system. For simplicity, a direct differentiator s is used to model the rate constraint, as in (Horowitz, 1984). Otherwise, a dynamic feedback loop may be used to model the rate constraint,
1. INTRODUCTION
In this paper the global stability problem for QFT designs for linear systems with rate and amplitude actuator saturation is analyzed. This problem is also discussed in (Hui, 2001), where a 3DoF scheme for the case in which there not exist plant uncertainty is proposed. However, the frequency domain design method proposed in (Hui, 2001) does not guarantee the global stability of system because a relaxed form of a sufficient condition (Freudenberg and Looze, 1988) very conservative to ensure global stability is used. In (Wu and Jayasuriya, 2001) the problem is discussed for the uncertain case, however global stability aspects are not considered. In relation to the placement of the amplitude saturation with respect to the rate saturation, two cases may be considered: when the control signal is saturated in rate before in amplitude (case considered in (Horowitz, 1984),(Liao and Horowitz, 1986), (Hui, 2001), (Wu and Jaya91
templates. In particular, X°(jw) = p°(jw)G(jw) represents the nominal point in the ni-template for frequency w. The ni-template and the I-template sets will allow us to deal with the uncertainty in an explicit form in the process design for the shaping of anti-windup compensator {Ha, H v }, in the second step of classical anti-windup paradigm, in which the non-linear operation mode is taken into account. Note that the shape of the nI-template is the same as the corresponding I-template, only the placement in NP is different. So that, computational effort is not required to computation of £fnl(W) if £f1(W) has been previously computed to design F and G compensators with QFT.
Fig. 1. A 4DoF control scheme with amplitude and rate actuator saturation. as in (Tyan and Bernstein, 1997). The paper is organized as follows. In section 2 some important definitions are introduced. In section 3 a set of results is proposed to assure the global stability of system in figure 1. In section 4 a design method is proposed to cope with the plant uncertainty. In section 5 an illustrative example is sho~n, and finally some conclusions are outlined in section 6. All proofs of the proposed results can be found in (Moreno, 2003).
Definition 3. For each frequency w, a boundary
b(w) is defined with respect to four vertical lines Plo (w), Pro (w), PI" (W) and Pr" (w), as the boundary with a region in NP so that if X°(jw) lies within
2. PRELIMINARIES
that zone, then one condition is satisfied (condition depending the context in which the boundary is used) for all plants P in p.
In figure 1 the linear block P represents a SISO LT! stable plant with transfer function P( s) belonging to a set of plants p. The compensator consists of four blocks: F(s), the precompensator; G(s), the feedback compensator, and {Ha(s), Hv(s)}, the anti-windup compensator. Assuming ideal saturation, the saturating elements outputs, Sat a and Sat v in figure 1, are given by the function
y(t) =
x(t) if x(t) E [min, max] min if x(t) < min { max if x(t) > max
Definition 4. For each frequency w functions br(w) and bl(w) are defined as
(1) so that
where x(t) = xa(t) and y(t) = Ya(t) for the Sat a element, and where x(t) = xv(t) and y(t) = yv(t) for the Sat v element, with xa(t) and xv(t) the Sat a and the Sat v inputs respectively, and with Ya(t) and yv(t) the Sat a and the Satv outputs. Amplitude saturation limits min = Y1nin and max = Y1nax, and rate saturation limits min = V 1n in and max = V 1nax are assumed. In the following, some definitions to be used afterwards are introduced.
(4) The equation (2) defines an allowed zone between
Plo(w) and Pr" (w) degrees, and (3) defines another allowed region between PI" (w) and Pro (w) degrees. Figure 4(a) shows an example of this type of boundary.
Definition 1. A ni-template (template for nonlinear mode) for each frequency w, £fnl(w), is defined as a set of complex value given by £fnl(w) ~
{P(jw)G(jw) : P E
3. STABILITY ANALYSIS FOR COMPENSATED SYSTEM
d.
In order to analyze the system stability is more convenient to transform the system in figure 1 to an equivalent system (from the stability point of view), consisting in a feedback interconnection of a linear block, including all linear dynamics present in the loop, and a non-linear block representing the saturating elements Sat a and Sat v . Considering signals Yv, Ya and r in figure 1 as system inputs, in order to compute Xv and X a as a function of them, and using ii to note the Laplace transform for a time signal a,
Definition 2. A I-template (template for linear mode) for each frequency w, £f1(W), is defined as a set of complex values given by £fl(w) ~ {P(jw) :
PEp}. Both of them are considered in the Nichols plane (NP), where complex numbers are represented as magnitude in decibels vs phase in degrees (generally in the [-360°,0°] range). For a nominal pO E p, a nominal point is obtained for both types of
92
Proposition 6, If conditions (i)-(iv) in Proposition 5 and the following conditions are satisfied
"
~ .
A..
all".
,
(v) U(G)i > U(P)i + 1, and (vi) U(G)i + 1 = U(Hv) and U(G)i + 1 = "(Ha), U(G)i + 1 = U(Hv) Y U(G)i = U(Ha), U(G)i + 1 = U(Ha) y U(G)i = U(Hv).
"
Fig. 2. Equivalent system with matrix blocks.
[ ~v] [1:~~V 1-:~~V] [~v] + =
Xa
~Ya s(1 + Ha) 1 + Ha
then the stability of K and the stability of Rare assured, but if the system in figure 1 is stable then an offset will appear in the output.
1
+[1
(5)
;~" i ~ K [t 1+ Ri ]
In the following, a version of the result based in robust stability with multipliers developed in (Moreno,2003) (using a previous result in (Zames and Falb, 1968» for SISO systems, to deal with amplitude saturation, is proposed to deal with simultaneous amplitude and rate saturation.
In figure 2 the resulting scheme is shown, where K and R blocks are given by equation (5) and with SATS non-linear block given by
SATS = [sat v
o
0 ] Sat a
(6)
Theorem 7. For the scheme of figure 2, if
A necessary condition for the system stability is that R in equation (5) must be stable. The stability for a transfer functions matrix given in (Skogestad and Postlethwaite, 1996) is used. The notation tt(')i is introduced to note the number of integrators in a transfer function. Defining functions Fv and Fa given by Fv="'--~
1 1 +sHv
(7)
F =_1_
(8)
a
I+H a
(i) The non-linearity SATS is diagonal, odd, monotone nondecreasing and belonging to slope
[O,lj, (ii) The linear part K(s) is stable, (iii) There exists some multiplier Z(s) = diag( Zl(S),Z2(S» (possibly non causal) and X = diag(X 1 , X 2) > 0, so that (iii-a) The impulse response Zi(t) for Zi(S) satisfies that IIzi(t)lll < Xi, Le
1:
00
and using equation (5) for matrix K, then K
=
[1 -
F v -SPGFv] Fa/s 1- Fa
IZi(t) Idt < Xi, for i
= 1,2
(iii-b) In frequency domain, for all frequency wand for some { > 0,
(9)
(X - Z(jw»(K(jw) + 1)+ + (K*(jw) + l)(X - Z*(jw»
is obtained. The matrix K stability, including the linear dynamics of system in figure 1, is key to apply different frequency domain stability criteria. The next Proposition sets necessary conditions to assure the stability for Kin (9), using the previous stability definition.
~
d
(10)
(iv) If SATS is even, then in addition it must be satisfied that
Zi(t)
~
0, Vt E (-00,00) with i
= 1,2.
then the interconnection is stable.
Proposition 5. If the following is satisfied
In the following Proposition the Theorem 7 is adapted to the case handled in this paper.
(i) U(P)i:S 1 VP E p, (ii) The number of zeros in origin for function Fa, given by (8), is greater or equal than one, (iii) All of poles of function Fa in (8) belong to open LHP (Left Half Plane excluding the origin), (iv) All of poles of function F v in (7) belong to open LHP, (v) U(G)i = U(P)i + 1 VP E p, (vi) U(G)i = U(Hv)i, and (vii) U(G)i = tt(Ha};.
Proposition 8, If there exist functions Fv and Fa (given by (7) and (8) respectively), and multipliers Zl and Z2, so that the next conditions are satisfied
(i) If Zl(t) and Z2(t) are the impulse response of Zl (s) and Z2 (s) respectively +00 ;
-00
;
-00
!zl(t)ldt < 1
+00
then the stability of K in (9) and the stability of R in (5) is assured, and if the system in figure 2 is stable then no offset will appear in the system output.
I Z2(t)!dt
<1
(ii) Conditions in Proposition 5 or conditions in Proposition 6 are satisfied 93
(Hi) -Fv(jw) "I- -2 + ja/(l - Zl(jW») "la E R,Vw > 0 (iv) -Fa(jw) "I- -2 + ja/(l - Z2(jW») "la E IR,Vw > 0 (v) Re( -P(jw)G(jw)Fv (jw)jw)Re(2 - Zl (jw) Z2(jW» 2: Im(-P(jw)G(jw)Fv (jw)jw)Im(2Zl(jW) - Z2(jW» Vw > 0 VP E P (vi) Re(Fa (jw)/(jw»Re(2 - Zl (jw) - Z2(jW» 2: Im(Fa(jw)/(jw»Im(2-Z1(jW)-Z2(jw» Vw >
.....1
Fig. 3. Boundaries over L°(jw) for frequency w in order to guarantee absolute stability for system without saturating elements.
o (vii) If SATS is even, then in addition the following condition must be satisfied
Using Z(s) = 0 a version of multivariable Circle Criterion is obtained, which is enunciated in the next Corollary.
Zl(t),Z2(t) 2: 0, "It E (-00,00) then the system in figure 2, with K given by (9), R given by (5) and SATS given by (6), is stable.
Corollary 10. If there exist functions Fa and Fv (given by (8) and (7) respectively) so that the following conditions are satisfied
4. CONTROL SYSTEM DESIGN
(i) Conditions in Proposition 5 or conditions in In this section a design method is proposed, taking the uncertainty of the plant into account, to choose compensators Ha and H v in figure 1, supposing that F and G have been previously designed using QFT, so that the uncertainty has been taken into account in the design process but only for the linear operation mode. In the following Corollary a result easier to check than in Proposition 8 is established.
Proposition 6 are satisfied, (ii) Re(Fv(jw» < 2 Vw > 0 (Hi) Re(Fa(jw» < 2 Vw > 0 (iv) ~P,w,w- ,w+ with w > O,w- < w < w+ and PEp: Im(P(jw-)G(jw-)Fv(jw-» "Io and Im(P(jw+)G(jw+)Fv(jw+» "I- 0 and Im(P(jw)G(jw)Fv(jw» = 0 (v) ~,w- ,w+ with w > 0 and w- < w < w+ : Im(Fa(jw-» "I- 0 and Im(Fa(jw+» "Io and Im(Fa(jw» = 0
Corollary 9. If there exist functions Fa and Fv (given by (8) and (7) respectively), and a multiplier Z (s) so that the following conditions are satisfied
then the system in figure 2, with K given by (9), R given by (5) and SATS given by (6), is stable.
(i) If z(t) is the impulse response of Z(s) +00
1
-00
Iz(t)ldt < 1,
(ii) Conditions in Proposition 5 or conditions in Proposition 6 are satisfied, (Hi) -Fv(jw) "I- -2 + ja/(1 - Z(jw» "la E R,Vw > 0 (iv) -Fa(jw) "I- -2 + ja/(1 - Z(jw» "la E IR,Vw > 0 (v) ~a,P,w,w-,w+ with w > O,w- < w < w+,a E IR and PEp: P(jw-)G(jw-)Fv(jw-) "Ia/(l- Z(jw-» and P(jw+)G(jw+)Fv(jw+) "Ia/(I-Z(jw+» and P(jw)G(jw)Fv(jw) = a/(IZ(jw» (vi) ~a,w,w- ,w+ with w > O,w- < w < w+ and a E IR: Fa(jw-) "I- a/(I-Z(jw-» and Fa (jw+) "I- a/(1 - Z(jw+» and Fa(jw) = a/(1 - Z(jw» (vii) If SATS is even, then in addition the following must be satisfied
If H v = 0 is supposed (the rate saturation effect is not compensated) and Ha(s) = l/s, a restriction over G compensator is obtained (which must be incorporated to the set of restrictions used in the loop shaping stage of QFT for the linear operation mode). This is established in the following Corollary.
Corollary 11. If the following conditions are satisfied
(i) H v = 0, (ii) Ha(s) = l/s, (iii) U(P); = 0 VP E p, (iv) ~(G); = 1, and (v) The system, in linear operation mode (without considering saturating elements), is absolutely stable for all plant P in p,
then system in figure 2, with K given by (9), R given by (5) and SATS given by (6), is stable. The result in Corollary 11 will provide a set of constraints over the phase of function L°(jw) similar to shown in figure 3. So that simply this set of boundaries needs to be added to the set of
Z(t) 2: 0, "It E (-00,00) then the system in figure 2, with K given by (9), R given by (5) and SATS given by (6), is stable.
94
P(s) = 0.2 (s2 +26W1s+w?) (S2 +26W1s+w?) s2 + 26w2S + w~ s2 + 26w2S + w~ (12) and 6 E [0.5,O.6J. In the linear mode operation the system is controlled using a Proportional-Integral law (PI) given by with
(a) Circle Criterion
(b) Multipliers
= 0.2115, W2 = 0.0473, ~l E [0.2,OAJ
{F(s),C(s)}
Fig. 4. Boundary over X°(jw) for frequency w in order to assure the global stability of system.
= {I, 100S;10
4 }
(13)
and the nominal plant po is given by 6 = 0.2 and b = 0.5. Amplitude saturation limits are assumed to be Ymax = 0.5 and Ymin = -0.5, and rate saturation limits are given by V max = 0.1 and Vmin = -0.1. In this example the set of designing frequencies W = {O.OI, 0.04, 0.2, 0.35} rad/s is used. Using the control law in equation (13), Ha(s) = l/s and Hv(s) = 0, it may be probed that the system is absolutely stable as a direct application of result in Corollary 11. Conditions (i)-(iv) in Corollary 11 are satisfied. In figure 5(a) the Nichols diagram of function X°(jw) = L°(jw) (with D = 1) is shown, as well as ni-templates and boundaries bl(w) for each frequency w in W. As it may be observed, function X°(jw) satisfies all restrictions. The global stability of the system is assured for all values of uncertain parameters 6 and 6 in (12). If in place of using the control law in (13) for the linear mode operation, the following is used (as it is proposed in (Doyle et al., 1987»
boundaries derived from specifications for linear operation mode, in the loop shaping stage in QFT, which will assure the closed loop system stability for all plants P in p, in presence of rate and amplitude saturation elements. Note that the Corollary 9 states less conservative conditions than Corollaries 10 and 11, because an additional degree of freedom is allowed, the multiplier Z (s), to assure the system stability. Condition (iii) in Corollary 9 allows us to rotate the not allowed locus for Fv function in condition (ii) in Corollary 10, a value given by 1 - Z(jw) for each frequency w. The same occurs with conditions (iv) and (vi) in Corollary 9 and conditions (iii) and (v) in Corollary 10 for the case of Fa, and with condition (v) in Corollary 9 and condition (iv) in Corollary 10. Let the function X given by
X(s) =P(s)C(s)Fv(s) = =P(s)C(s)D(s) with PEp
Wl
(11)
{F(s),C(s)} =
with function Fv given by (7). It is supposed that {F, C} compensator was designed in a previous step. Note that shifting of ni-template for frequency win NP is equivalent to add ID(jw)1 decibels and Angle(D(jw» degrees to each point in ~1(W), or equivalently to the nominal X°(jw), assuming this as representative of the rest of points. So the introduction of poles and zeros in D in (11) is equivalent to shift the ni-templates. When D is designed the computation of H v is straightforward from (7). Then zones in NP must be defined so that if XO lies within them, then stability may be concluded. These zones can be defined using four vertical lines as shown in figure 4(a). For a more detailed example see section 5.
{I, 100Ss+ 1O}
(14)
and Ha(s) = l/s and Hv(s) = 0 are taken, then the function X°(jw) with D(s) = 1 does not fulfill the constraints, so that it is necessary to use an adequate compensator H v . A possible choice for Fv is Fv(s) = 2.5s + 1 (15) 4.975s + 1 which introduces an increase in the phase approximately of 20 degrees around w = 0.3 rad/s. Figure 5(b) shows boundaries bl(w) and ni-templates ~nl(W) for each frequency w in W, shifted by the shaping D = Fv , and function X°(jw) which, as it may be observed, fulfills all constraints. So, as Fv satisfies conditions (i) and (ii) in Corollary 10, the global stability of system is assured for all values of parameters 6 and 6 in (12).
Using the result in Corollary 9 the set of four vertical lines may be shifted in the NP using a multiplier Z (s). In figure 4(b) a typical boundary b(w) using this result is shown. For a more detailed example see section 5.
Finally, the global stability for design given by the control law for linear operation mode in (14) and Hv(s) = 0 (supposing that Ha(s) = l/s) may be probed too. Choosing the multiplier Z (s) = 1 Fv(s) with Fv given by (15), a modification in the location of boundaries is introduced, such that the system for F v = 1 (Hv = 0) satisfies all constraints. Conditions (i)-(iv) and (v) in Corollary 9 are satisfied. Restrictions on phase of X°(jw) are fulfilled too, as shown in figure 5(c). So, the global stability of system may be concluded. Note that
5. AN ILLUSTRATIVE EXAMPLE The plant considered for this example has been previously used in (Doyle et al., 1987), but uncertainty has been introduced in two parameters in order to study the proposed method. 95
grants QUI99-0663-C02-02, DPI2000-1218-C0403, DPI2001-2380-C02-02, and DPI2002-04375C03-03.
..1:
J:
:j!
;
"
/--
-*:
4:
'1.:
REFERENCES
0,
(a) D(s)
=1
i:
(b) D(s)
= Fv
Doyle, J. C., R. S. Smith and D. F. Enns (1987). Control of plants with input saturation nonlinearities. In: Proceedings of the 1987 American Control Conference. Vol. 145. Minneapolis. pp. 1034-1039. Freudenberg, J. S. and D. P. Looze (1988). Frequency Domain Properties of Scalar and Multivariable Feedback Systems. Springer Verlag. Berlin. Horowitz, I. M. (1984). Feedback systems with rate and amplitude limiting. International Journal of Control 40, 1215-1229. HUi, K. (2001). Design of rate and amplitude saturation compensators using three degrees of freedom controllers. International Journal of Control 74, 91-1Ol. Kapasouris, P. and M. Athans (1990). Control systems with rate and magnitude saturation for neutrally stable open loop systems. In: Proceedings of 29th IEEE Conference in Decision and Control. pp. 3404-3409. Liao, Y. K. and I. M. Horowitz (1986). Unstable uncertain plants with rate and amplitude saturations. International Journal of Control 44, 1147-1159. Moreno, J. C. (2003). Robust control of systems with input constraints. PhD thesis. Dept. of Computer Science and Systems. University of Murcia. Moreno, J. C., A. Baiios and M. Berenguel (2002). Robust control for linear systems with actuator saturation using qft. Submitted. Skogestad, S. and I. Postlethwaite (1996). Multivariable Feedback Control. Wiley. New York. Tyan, F. and D. S. Bernstein (1997). Dynamic output feedback compensation for linear systems with independent amplitude and rate saturations. International Journal of Control 67,89-116. Wu, W. and S. Jayasuriya (2001). A qft design methodology for feedback systems with input rate or amplitude and rate saturation. In: Proceedings of the 2001 American Control Conference. Arlington, VA. pp. 376-383. Zames, G. and P. L. Falb (1968). Stability conditions for systems with monotone and slope-restricted nonlinearities. SIAM Journal 6(1),89-108.
~
...•.• . :
~t
-':
,
(c) D(s) tiplier
= 1 with mul-
Fig. 5. ~nl(W) with w in Wand nominal X°(jw). in this example the value for 1- Z (s) is the same as previous Fv , so that we could think that the use of Fv or the multiplier is totally equivalent. However there are two important differences. In first place, the fact of probe the stability of system without using Fv implies that H v = 0, and in the second place, using Fv implies an advantage over the use of the multiplier Z (s) because there are no restrictions over the L 1 - norm of impulse response of Fv and there are restrictions over L 1 - norm of impulse response of Z. Furthermore, the use of the multiplier does not imply to exclude the use of Fv , both of them may be used together (as it is deduced, for example, in Corollary 10), providing the proposed method with great flexibility.
6. CONCLUSIONS In this paper the problem derived from the presence of rate and amplitude saturation in the control system with an uncertain plant is analyzed from a stability point of view. To deal with the problem a 4DoF control scheme, two (F and G) designed in a first step with QFT so that the uncertainty is taken into account for the linear operation mode, and the other two (Ha and Hv ) designed in a second step to deal with the non-linear operation mode. Taking as a basis the Zames-Falb result about stability with multipliers, a set of results and an algorithm to design Ha and Hv for an uncertain plant assuring the global stability of system in the presence of rate and amplitude saturation are proposed.
ACKNOWLEDGEMENTS The authors would like to thank the Spanish Comision Interministerial de Ciencia y Tecnologfa (CICYT) for partially funding this work under 96