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Design of robust control systems with time-domain specifications
ControlEng.Practice,Vol. 3, No. 3, pp. 365-372, 1995
Pergamon
Copyright© 1995 Elsevier Science Ltd Printed in Great Britain. All rights rosa.red 0967-0661/95 $9.50 + 0.00 0967-0661(95)00008-9
DESIGN OF ROBUST CONTROL SYSTEMS WITH TIME-DOMAIN SPECIFICATIONS S. Engell Lehrstuhlfiir Anlagensteuerungstechnik, FachbereichChemietechnik, UniverMtdtDortmund, D-44221 Dortmund, Germany
(Received August 1993; in final form September 1994)
Abstract. A novel derivation of an IHoo-Optimization problem from realistic design specifications is presented in this paper. The key idea is that three different independent design specifications are included which specify robustness, sensitivity, and transient response. The three specifications are pasted together by suitable switching functions. In the scalar case, the frequency response of the closed-loop system is restricted by frequencydependent disks in the complex plane with complex center points. The use of the approach is demonstrated for a real-world design example with actuator limitations and unstable plant dynamics. Key Words. Robust control; control systems design; IH®-optimization; control of unstable plants
"look good", the practical advantage of optimal control approaches may not be too large, because experimentation with different controllers is only replaced by experimentation with different weighting functions. It may even, in certain cases, be difficult to match the result and the efficiency of the classical techniques by optimization-based methods.
I. I N T R O D U C T I O N Standard optimization techniques ( I ~ - and IH.optimization) require the specification of weighting functions in a fixed performance criterion. The "optimization" is in fact a predetermined mapping from the set of plants and weighting functions to the set of all stabilizing controllers. Choosing the weights means choosing a controller and thus choosing a closed-loop system.
The example of temperature regulation in a nuclear reactor illustrates this statement: The plant model is 6th order with one unstable pole at + 0,0011, two rhp zeros at 6.14 a:j 5.96. The actuated input has to satisfy a rate limitation which limits the attainable bandwidth (see Appendix). This control problem was first discribed by Postlethwaite et al. (1987) where IH~o-mixed sensitivity minimization was applied. To use this method, the following steps must be performed:
The weighting functions however are at best partially related to the original engineering specifications. No analytically tractable optimization problem so far can represent the relevant constraints and goals in a real design problem. Optimality with respect even to reasonably chosen weighting functions may be neither sufficient nor necessary for a good design.
1) map the rate limitation into a weighting function for the complementary sensitivity function 2) select a weighting function for the disturbance-tooutput transfer function with a free parameter that can be optimized within the range allowed by the first weighting
Compared to the classical (frequency domain) design methods where the designer manipulates the openloop system in a series of trial-and-error steps until both the frequency response and the time response
365
366
S. Engell
3) compute the optimal controller 4) investigate whether its order can be reduced. Step 3 is solved by algorithms resulting from IH~ooptimal control theory. For step 4, there are standard methods available for the removal of irrelevant parts of the controller dynamics, e.g. truncation of balanced realizations or I-Iankel-norm minimization. If one wants to go beyond this, some experimentation may be needed because a specific frequency-dependent weighting is indispensable (Engell and Kerker, 1989). Step 1 is straightforward if the order of the weighting function does not matter. A high order however makes steps 3 and 4 computationally more difficult. Otherwise, some engineering judgement is necessary. Step 2 is the least systematic one. One may use a very simple weighting function, as it is frequently done, but why should this be the "optimal" choice? Postlethwaite et. al. (1987) used first-order weighting functions, the sensitivity was minimized, and the resulting controller was finally reduced from order 5 to order 2 with almost no effect on its response. How would one approach the same problem using classical design techniques? First the structure of a stabilizing controller has to be determined. This clearly requires some knowledge and insight, and is facilitated by the use of appropriate CAD-tools. From inspection of root loci or Bode plots, it can be concluded that an integral controller with negative gain will stabilize the loop. After mapping the rate limitation into a bound on the bandwidth as above (step 1), it can be seen that with this controller the maximal possible bandwidth can be achieved, and the gain is then increased until the rate limitation is reached. The results of the two approaches are compared in Fig. 1. Most control engineers will probably agree that the second approach did not result in an inferior performance compared to the first one, but obtained this with a simpler controller, and, with some experience, less design effort. In both cases, one would like the controlled variable to converge faster to its steady-state value. In the classical approach, it is clear what to do: insert an additional lag compensator. It turns out however, that this inevitably produces more overshoot. So here the method reaches its limitations: it cannot be decided without extensive trials whether a more complex controller would allow one to obtain both small overshoot and rapid convergence. (In fact, it can be shown that there is an inevitable trade-off under the given bandwidth limitation, see Appendix 1.) But it is also not obvious how to proceed further using the mixed-sensitivity minimization approach, because the shape of the transient response is not
directly related to the choice of the weighting functions.
r.~/I
t.2
*
Log
.......................................
1
O.II
IH,o mixed sensitivity optimization
I
0.4
O.= 0
Fig. 1. Step responses obtained by IH.o-mixed sensitivity minimization and by classical design techniques This contribution describes a novel approach to controller design which uses IH®-optimization but is more closely related to practical design goals than the standard approach because the desired transient behaviour enters directly into the cost function. The goal is to integrate the benefits of IH**-optimization with the easy control of the transient behavior provided by classical design techniques. 2. D E S I G N S P E C I F I C A T I O N S In this section a minimal set of specifications is stated which are encountered in any realistic design problem, and these specifications are mapped into frequency-domain constraints. This is the basis for the derivation of a modified IH®-optimization problem which encompasses this set of specifications in a transparent way. Consider the standard feedback configuration shown in Fig. 2, where V(s) = _~s)W(s) +_S(s)_Z(s). (1)
W(s=) ~
Z(s)
Fig. 2. Standard control loop The starting point here is the observation that in almost all practical cases, there is a hierarchy of design specifications: 1. Hard constraints due to plant uncertainty, measurement noise, and actuator limitations. The
Design of Robust Control Systems
simplest representation of these constraints is a norm bound on the reference-to-output or complementary sensitivity transfer function or transfer matrix: ]T(jco)
(2a)
367
is equivalent to finding the minimal value of v for which the frequency response SOco) lies, for each value of co, in a disk with radius r(m) and center c(m). c(co) and r(r~) are related to Wl(jm) and W2Om)
by W2(J°~) 2
or
VT(jm)_~jo~)WT(j~ ) < I
(5a)
w,(jm)2+ w (jm) 2
(2b)
where H is the maximal singular value. r(~)= c(~
One well-known special case of (2a) is
W2.jto. 2 +c(co)-I
.
(5b)
T(jco) <12(m ) , (Doyle and Stein, 1981) where 1m is a bound on the multiplicative plant uncertainty. (2a,b) usually implies a specification of high-frequency roll-off. 2. Feedback control is always introduced to counteract unmeasured disturbances and plant uncertainties. This means small values of the sensitivity transfer function or transfer matrix over some set of frequencies and not too large values elsewhere, hence
S(jm) < as(m)
(3a)
or
So the centers of the circles move from 0 to I on the real axis as ]W I [decreases and [W2[ increases, ff [W I [>>[ W2 [, (4) is approximately equivalent to
s(j )
(3b)
(3a,b) in particular implies a certain bandwidth of the control loops and the asymptotic behavior as o~ --, O. 3. In general, control systems are judged upon by testing time responses, particularly step responses. Good damping, reasonable peak values and fast convergence to the steady-state value are general quality criteria. The step response is determined, for the most part, by the behavior of the control loops near the gain crossover frequencies. As the previous example shows, reasonable specifications of Ir(jco) l at high frequencies and of I S(jco) I at low frequencies alone do not ~ t c c a good transient response. As far as the reaction to reference steps is concerned, the desired behaviour can always be achieved by introducing a shaping filter in front of the loop, i.e. by using a 2-degree-of-freedom structure. The reaction to disturbances, e.g. load changes, however is still determined by the loop transfer functions S and T. How are these specifications reflected in mixed sensitivity minimization? It can be shown that the minimization of v in
v = sup{ WI(jm)S(j~) 2 + W:(jm)T(jm) : }
(4)
(6a)
and ff IW: I >>1Wl I, (4) is approximately equivalent to T(j~) <,T(jro)= ~r~. W:(jc0) -1.
(6b)
I>>lwel
Usually, IW 1 will hold as co --} 0 to antee good disturbance rejection whereas [>>[Wl[will hold as co -} ~ to ensure robustness against unmodelled or unknown dynamics. So it is relatively easy to represent the specifications 1 and 2 in the form (4). [
_Vs(jm)_S(j~)Ws (jco) <1.
w,(jm)-',
t
t
By minimizing (4), the weighted fit of S0~) and a real reference function which increases from 0 to 1 is optimized. This implies that there can be little control of the phase of the closed-loop system because S(j~) can never be close to such a real function for all frequencies. The weights influence both the reference function (the centers of the circles) and the size of the circles in a not too transparent manner. If standard simple weighting functions are used, the behaviour in the intermediate frequency range and thus the step response is not well controlled, because the resulting circles there are large. This argument holds as well in the multivariable case where not only the transient behaviour of the main loops but also the coupling terms may be of interest. In order to achieve better control of the transient behaviour near gain crossover the open-loop frequency response should be close to that of a reference model which provides the desired closedloop step response. From this, the third specification T(jco)- M(j~) < ~M(~)
(7a)
in the scalar case o r VM(j~)[_T(j~)- M(j~)]WM(jm) [ < 1 in the multivariable case results.
(7o)
368
S. Engell (9)
In (7a,b), the specification of the transient behaviour is in terms of T(jCO) which is the response to a reference step in the 1-dof structure. In a 2-dof structure, 1-T defines the response to disturbances at the output, Gp . ( l - T ) the response to disturbances at the input of the plant. A model for T is used here because the relation of the transfer function to the step response is well known from the classical design rules and their extensions (Engell, 1991). For a scalar plant which is minimum-phase except for a pure delay and one dominating unstable pole or one dominating rhp zero, a second-order system is usually sufficient as a reference model because additional requirements can be incorporated into (2) and (3). Second-order models can easily be parametrized from time-domain or time and frequency domain specifications. In the multivariable case, the choice of M is more difficult. Often, one will specify a diagonal transfer matrix M using scalar techniques for the specification of the diagonal elements, and diagonal weighting matrices to account for the different bandwidths. By the three specifications (2), (3), (7), all basic design requirements are represented in a a uniform manner, ff the specifications (2) and (3) are fixed from the practical problem, the designer will try to optimize the transient behavior within these constraints, e.g. by choosing M(s) such that the bandwidth roughly agrees with the constraints imposed by (2) and (3) and minimizing ~ . Alternatively, the effect of smaller values of es at low frequencies on the attainable fit to the reference model can be investigated. In the SISO case, the set of specifications (2a), (3a), (7a), constitutes a 3-disk-problem which is not analytically tractable. It may also not have a solution at all, ff the three disks do not overlap at each frequency, i.e. the specifications are not compatible. However, it frequently suffices to meet the specifications in different frequency ranges. (2) will be imposed at high frequencies, (3) at low frequencies, and (7) in the intermediate range. So the design problem in the scalar case is formulated as: Find a stabilizing controller such that
S(jCO)-c(CO) < r(CO) with
VCO
fi
c(CO)= -M(jCO)
ICOl COb
Ob
ICOl---COz and
I
(10)
ICOl-< COb
s(CO)
r(CO)= M(CO)
COb< ICOl< CO,"
/
{F--'T(CO)
ICOl CO,
Note that c(CO) is complex. The problem of the compatibility of the specifications is then reduced to the compatibility of (la) and (7a) at COband (3a) and (7a) at COz.In general, one should choose COb,COzsuch that M(jCOb) = ~S(COb),
M(jCOz) = ~;T(COz)
(11)
in order to have a smooth transition from one specification to the other. In the multivariable case, a similar representation with a piecewise defined center function matrix and right and left weighting matrices is V(jCO)[_C(jCO)-_S(jCO)]W(jCO)[< 1
(12)
3. T R A N S F O R M A T I O N I N T O A N I H o o - O P T I M I Z A T I O N PROBLEM The existence of a stabilizing controller which satisfies (8) or (11) can checked to any desired precision by the introduction of rational approximations to c(CO)and r(CO)or V, C, and W. In the scalar case, the approximation is specified as follows: outside the intervals ~ s _ < ~_o___< 1 COb ~ S
o_~_<
'
~T~
1 ~T
'
c(CO) and r(CO) are approximated with prescribed maximal approximation error -
inside these intervals, the approximation error is bounded and at most as large as ~ / 2 times the difference of the two functions which are pasted together at COb or COz.
(8)
So the approximation introduces significant (but still bounded) errors only in small intervals around COb, coz, and arbitrarily small errors elsewhere. The relative values of the errors outside the transition ranges and the size of the transition range are related for a given order of the approximation, but both can
Design of Robust Control Systems
be made as small as desired by increasing the order of the approximation. Suitable approximations can then be constructed using the so-called Zolotarev-functions (Piloty, 1954). K(s/%) denotes the n-th order Zolotarevfunction with comer frequency col. It satisfies, among all functions of the same order, [K(jco')t < $
for ~ 1 < ~
(13a)
IK(jo')l>8-'
for ~']>l/~f~
(13b)
for the maximal possible value of I] for given 8 or, vice versa, the minimal value of 8 for given [~. The functions, first obtained by Zolotarev have the form -jz ]2I i=1 1+ b~z2
(n odd)
"= l+b2z2
(18)
where Es(s), E~s), Er(s ) are rational functions with modulus es(CO), eM(CO), ~(0)) on the jco-axis, and { }+ denotes the stable minimum phase transfer function having the same modulus on the jo-axis. After Vc(s) and Vr(S) are chosen, it can be checked whether
plls(Jo)- vo(jo)l-qv,(Jo)l} o
(19)
~0
(14a)
(n even)
(14b)
(z=co/%). The parameters b i depend on ~ or 8 (Caner, 1940; Pilot),, 1954). K(z) increases monotonically in the transition interval and has modulus 1 at z = 1. From these functions, rational approximations to the unit-step function on the j~axis (s' = s / % ) are constructed:
(_i)vs,kKn(/~ ]
This problem can be solved using any of the available techniques, e.g. Pick-interpolation (Pick, 1916; Engell, 1988), polynomial methods (Meinsma and Kwakernaak, 1991) or state-space algorithms. The minimal value of ,c can in fact be computed without iteration. The minimization of r in (19) is identical with the solution of the "generalized I ~ optimal control problem" which was introduced independently by Grimble (1989) because of computational advantages but not discussed with respect to the choice of weighting functions.
(15)
P(')= 1+(_l)~,(s,k)k Kn(~) Apart from the factors s'k these are the well-known elliptichigh-pass filterfunctions.They satisfy P(jco')
Vr(s) = (Es(s) [1 - Pb(~b/ + EM(S)Pb(~-b)[1 - Pz(~z/] +
can be satisfied for ~c < 1 with a stabilizing controller. Vr is stable, but Vc in general is not.
(n-1-'----~)_(b ~ + Z2)
='
369
for k o ' l < ~
(16a)
I - P ( j o ' ) <~'1 k. 8 forlco'l>--1/f~. 1-8
(16b)
If k and n are both even or both odd, T is given by the condition that P must remain finite on the imaginary axis, and P(jco) is real. Otherwise, 7 can be chosen freely, and depending on this choice the Nyquist plot of P(jco)is a semicircle from (0, j0) to (I,j0) in the upper or in the lower half-plane.
4. DESIGN E X A M P L E The use of this IH®-optimization approach with 3 independent design goals is illustrated for the example discussed in Section 1. The hard constraint in this case is given by the actuator rate limitation. It can be mapped into the frequency domain specification
T oI=bI ,C o) The next step is to specify a model for the transient behaviour. Starting as simply as possible, the fact that the plant has an unstable pole and unstable zeros was ignored at first and a standard second order model with 10% overshoot was chosen and the bandwidth was roughly adapted to (20). The result was
The factors s'k are necessary to prescribe the convergence of the error for co - , 0 and co - , ~. The rational approximations V¢(s), Vr(S) finally are
Vc(s)=
Pb S 1 M ( S ) 1 (~bb) f-
Pz s
(~z~]
+Pz s
(~zz) (17)
MI(s ) -
1
1+ 23s + 400s 2
Now if under the constraint (20), a controller can be found which gives a system which is a good approximation problem would be solved. This can be using ~s,(O) = 1- M,(jo)
(21) stabilizing closed-loop of M 1, our checked by (22)
370
S. Engell
below rob and minimizing aM (a constant value was used here in general) The result was, however, that the convergence of the step response to the steadystate value is unsatisfactory (as before). So S0ro) must be made smaller at low frequencies. To achieve this,
rr,'k
1.2
1.0
0.1
0.1,
Es2(s) _ ks(S+COb)
(23) 0.4
S2
with k s such that used.
.......
¢$1
-.-*-.-
u=+ • 0.00,~1
--.o--.
u= • o . o o e J+ ¢sz
Hi
Es2(J%)= 1-Mr(job)was
0.2
0
was again minimized for different values of %. roz was chosen as 0.04 which is the frequency where ~r(O~) is equal to 1. The orders of the functions Pb, Pz were 7 and 8 with ~/~= 0.7. The resulting frequency responses for minimal values of eM are shown in Fig. 3, the corresponding step responses in Fig. 4. It turns out that the desired behavior cannot be matched for any reasonable value of %. This is due to the unstable pole of the plant which was not included as a zero in M v One may pick a value of rob as a reasonable compromise, e.g. rob= 0.004, and then simplify the resulting controller. The allowed disks and the optimal solution for this value are shown in fig. 5 as an illustration of the geometry of the 3-disk specification and the well-known behavior of the solution to he always on the edge of the specified region.
'
,~0,
'
,~0,
'
,,=,'
'
,+'=.
'
,o=,
Fig. 4. Step responses corresponding to Fig. 3
2
S(ja))
MI,~s2
Vc(jm) 1
+
0.5
~ P:.<:" . ..... :~ ,, : (/", ,/,," '-, ,,", ' / / : . "2-22; --.,'.'.. i ! ,i' ~/li -3 ,. \ ", ","-'-=";',','-/"/'
/4 // I/,?'
-0.5 The orders of the approximations must be relatively large to provide a rapid transition from one specification to the next. Otherwise, there is significant interaction and the goal to evaluate three independent specifications cannot be met. The exponents k must be such that the approximation error remains small compared to 6 s and %. The order of K(z) must at least be 3, but 5 is preferable. In Fig. 6, the moduli of Vc and Vr are shown together with those of S and 1 - M 1 for gs2 with rob = 0.004 as in Fig. 5. More details on the effect of the approximation can be found in (Engell and Oukhai, 1991).
"~1
-0.5
o25
d
i
125
2
Fig. 5. Prescribed regions and optimal frequency response for rob = 0.004
2O 0 -2O -40 -60:
IStjmll.lmlj=ll
d8 lO
. . . . . . . .
i
. . . . . . . .
i
-8(
. . . . . . . .
i
. . . . . . . .
j, `I'` 0J I+ ' ' `
1~
. . . . .
ISljul
.I(X •
O.
•
o
~
__._ -120 -140
-lO,
-160
-2o. "1 ~ 0 d 4
-3o,
-40.
-50.
-50
/
//
//
---,-.-
..... IVctju)l .......o.... IVr (ju)l
i
--x---
1
........
, 10.=
,
, ,,,,,
I1 - 141 ( j u } l
. . . . . . 104
,. . . . . . . . . . . 10.1
foal=
I
tOa
Fig. 6. V c and V r for Mr(s), ¢ob = 0.004
...... o;.ooo,r' ,
,//4:/ . . . .
w= = 0 . 0 0 4 ]
,' .......... "",/
'-",
•, g l / . , , t
it)";I
i
i
i i iiii
I
i0"I
. . . . . . . .
i
1'0"I
. . . . . . . .
rod/=
100
Fig. 3. [ S(jm)l for M 1 (s) and Ss) or SS2 f o r v a r i o u s values of rob, ~M minimized
To investigate more deeply whether a better performance is possible, the next step is to respect the unstable pole in the reference model by choosing it such that M(Pu)=l. The simplest reference model
Design of Robust Conti"ol Systems which achieves this is a second-order transfer function with a zero in the nominator. It was specified such that the gain at % is 1 to be compatible with the bandwidth limitation (20), and the rise-time was optimized for a desired overshoot of 10%.
2
S(j~)
-
1.5
371
-.
M2
vc(jco)
1
0.5 Using this reference model instead of M~ however gave unsatisfactory results. The resulting value of is so large that there is no resemblance of the step responses. This is due to the fact that above ~z, ST is much smaller than the gain of the reference model and therefore its phase cannot be matched below o~z because of the higher required roll-off above coz. Therefore the reference model was modified by adding an additional pole at 0.06 to obtain roughly the same roll-off as ~ . In order to get 10% overshoot, M 2(s) 1+ 116s (24) 1+ 127s+ 3524s 2 + 2 8 0 6 0 s 3 was chosen. Note that M2(Pu) does not have exactly the desired value at p,, because this would result in too much overshoot again and the goal was to explore whether this is inevitable or not. No additional attenuation at low frequencies was specified. Then the Optimal closed-loop system which minimizes the error between T and M.2 in the intermediate frequency range and satisfies the stability conditions has the step response shown in Fig. 7. The approximation of M 2 is very good (compare Figs. 8, 9). If more disturbance attenuation at low frequencies is asked for, the error between T and M 2 increases again. Essentially, the results then are the same as obtained with M r
0 '~" \"~.",'";'-:'.." ,:,/i!/ ..... ,,~..."............. ~..:,;y ,~,~:........... :.~,,,
-0.5 -I
-1.5
-2i
-1
6
i
2
Fig. 8. S(j~) and prescribed regions for minimal Using the proposed approach, a clear picture of the inevitable trade-offs was obtained for this example independent of a particular structure of the controller. The problem encountered using t h e classical frequency domain design technique was found not to be a result of that particular technique but a consequence of the plant dynamics, i. e. the unstable pole, and the specification (20). This can also be proven analytically (see Appendix 1). The specification of three independent performance measures allowed us to explore the limitations on the attainable performance in a precise manner. The example also shows that it may be difficult to get as good results as those obtained by purely interactive design by using standard optimization methods.
d8 20 0 .20 4O -6O 410
0. .12G
0.6
--x---
11 - M 2 ( j w } I
-140 / .ll
0.~
0.2
1o-,
I~
I~
d-,
Io~
1o,
I~ ,.¢ko,
Fig. 9.]Vc and]Vr forM2(s),G M minimized s~O. lO00m 1200.
14oo,
I~,
I~LOO. )~O(Om
Fig. 7. Step response obtained for M2(s)
All controllers can be reduced to order 2 using the technique described in (Engell, 1991) without loss of performance.
5. S U M M A R Y A novel formulation of the controller design problem in the framework of IH®-optimization was described. This formulation includes all three major design specifications: robustness, disturbance rejection, and
S. EngeH
372
transient response. The procedure can be extended to the multivariable case in a straight-forward manner by pasting the piecewise definied weighting matrices or center function matrices together as in the scalar case. However, to prevent an explosion of the order of the controller, the specifications (2b), (3b), and (7b) should be pasted together at the same frequencies for all elements of the matrices.
APPENDIX 1: Attainable step responses for unstable SISO plants Assume a SISO plant has an unstable pole at s = p in the rhp. Then for internal stability of the closed-loop system shown in Fig. 2, T(p) = 1
(m)
~gT(t)e-Ptdt = 1 0
where gT(t) is the impulse response of T. Integration by parts yields (A3)
0
As
~e-ptdt = p-l, 0 co
f[1-hT(t)] e-ptdt = 0.
Poles
Zeros
+.0011 -.0125 -.0727+/-j.0778 -1.044+/-j.610
-.0023 -.0129 -.207 -1.23 +6.14 +/-j 5.96
DC-gain: 41.49 Rate limitation: du/dt < .00222 for unit reference step
Acknowledgement: The author is very grateful to I. Postlethwaite and S.D. O'Young for the details of the example.
(A1)
is necessary. Let hT(t) be the step response for reference steps of the closed-loop system. Assume hT(0) = 0 (strictly proper plant) and hT(t) --> 1 for t oo (steady-state accuracy). From (A1),
~hT(t)e-Ptdt = p-I
APPENDIX 2: Data of the example (plant model)
(A4)
0
Formula (A4) restricts the attainable step responses: the integral of the control error, weighted by exp(pt) must vanish. As hT(t) is below 1 for small values of t, this area must be compensated by values above 1 for larger t, and this compensation is discounted as t -~ oo, so the step response cannot show an infinitesimally small error for a long time to satisfy (A4). If an unstable pole and actuator limitations are both present, the initial error integral of the step response is bounded from below, and hence there must be a positive control error the time integral of which is at least as large as the initial error integral, hence either large overshoot and rapid convergence to the steady-state value or small overshoot and slow convergence occur, as encoutered in the example.
REFERENCES Cauer, W. (1940). Bemerkung fiber eine Extremalaufgabe von E. Zolotareff. Zeitschr. f. Ang. Math. u. Mech., 20, 358. Doyle, J., and Stein, G. (1981). Multivariable Feedback Design: Concepts for a ClassicalModern Synsthesis. IEEE Tr. on Automatic Control 26, 4-16. Engell, S. (1988). Optimale lineare Regelung. Springer-Verlag, Berlin. Engell, S. (1991). Towards a Complete Frequency Domain Design Methodology. 1st IFAC Symposium on Design Methods of Control Systems, Preprints, 66-72, Ztirich. Engell, S., and Kerker, R. (1989). Controller Design by Approximation of Ideal Compensators. Proc.1989 American Control Conference, 18991900, Pittsburgh. Engell, S., and Oukhai, A. (1991). Model Matching Under Hard Constraints: A Case Study. Proc. 1st European Control Conference, 709-714, Grenoble. Grimble, M.J. (1989). Generalized IH~Multivariable Controllers. IEE Proceedings D, Vol. 136, 285-297. Meinsma, G., and Kwakernaak, H. (1991). IH~oOptimal Control and Behaviours. Proc. 1st European Control Conference, 1741-1746, Grenoble. Pick, G. (1916). Woer die Beschr~inkung analytischer Funktionen. Math. Ann., 77, 7-23. Piloty, H. (1954). Zolotareffsche rationale Funktionen. Zeitschr. f. Ang. Math. u. Mech., 34, 175-189. Postlethwaite, I., O'Young, S.D., Gu, G.-W., and Hope, J. (1987). IH~-Control Systems Design: a Critical Assessment. IFAC World Congress, Prepr. Vol. 8, 328-333, Miinchen.
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Copyright© 1995 Elsevier Science Ltd Printed in Great Britain. All rights rosa.red 0967-0661/95 $9.50 + 0.00 0967-0661(95)00008-9
DESIGN OF ROBUST CONTROL SYSTEMS WITH TIME-DOMAIN SPECIFICATIONS S. Engell Lehrstuhlfiir Anlagensteuerungstechnik, FachbereichChemietechnik, UniverMtdtDortmund, D-44221 Dortmund, Germany
(Received August 1993; in final form September 1994)
Abstract. A novel derivation of an IHoo-Optimization problem from realistic design specifications is presented in this paper. The key idea is that three different independent design specifications are included which specify robustness, sensitivity, and transient response. The three specifications are pasted together by suitable switching functions. In the scalar case, the frequency response of the closed-loop system is restricted by frequencydependent disks in the complex plane with complex center points. The use of the approach is demonstrated for a real-world design example with actuator limitations and unstable plant dynamics. Key Words. Robust control; control systems design; IH®-optimization; control of unstable plants
"look good", the practical advantage of optimal control approaches may not be too large, because experimentation with different controllers is only replaced by experimentation with different weighting functions. It may even, in certain cases, be difficult to match the result and the efficiency of the classical techniques by optimization-based methods.
I. I N T R O D U C T I O N Standard optimization techniques ( I ~ - and IH.optimization) require the specification of weighting functions in a fixed performance criterion. The "optimization" is in fact a predetermined mapping from the set of plants and weighting functions to the set of all stabilizing controllers. Choosing the weights means choosing a controller and thus choosing a closed-loop system.
The example of temperature regulation in a nuclear reactor illustrates this statement: The plant model is 6th order with one unstable pole at + 0,0011, two rhp zeros at 6.14 a:j 5.96. The actuated input has to satisfy a rate limitation which limits the attainable bandwidth (see Appendix). This control problem was first discribed by Postlethwaite et al. (1987) where IH~o-mixed sensitivity minimization was applied. To use this method, the following steps must be performed:
The weighting functions however are at best partially related to the original engineering specifications. No analytically tractable optimization problem so far can represent the relevant constraints and goals in a real design problem. Optimality with respect even to reasonably chosen weighting functions may be neither sufficient nor necessary for a good design.
1) map the rate limitation into a weighting function for the complementary sensitivity function 2) select a weighting function for the disturbance-tooutput transfer function with a free parameter that can be optimized within the range allowed by the first weighting
Compared to the classical (frequency domain) design methods where the designer manipulates the openloop system in a series of trial-and-error steps until both the frequency response and the time response
365
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S. Engell
3) compute the optimal controller 4) investigate whether its order can be reduced. Step 3 is solved by algorithms resulting from IH~ooptimal control theory. For step 4, there are standard methods available for the removal of irrelevant parts of the controller dynamics, e.g. truncation of balanced realizations or I-Iankel-norm minimization. If one wants to go beyond this, some experimentation may be needed because a specific frequency-dependent weighting is indispensable (Engell and Kerker, 1989). Step 1 is straightforward if the order of the weighting function does not matter. A high order however makes steps 3 and 4 computationally more difficult. Otherwise, some engineering judgement is necessary. Step 2 is the least systematic one. One may use a very simple weighting function, as it is frequently done, but why should this be the "optimal" choice? Postlethwaite et. al. (1987) used first-order weighting functions, the sensitivity was minimized, and the resulting controller was finally reduced from order 5 to order 2 with almost no effect on its response. How would one approach the same problem using classical design techniques? First the structure of a stabilizing controller has to be determined. This clearly requires some knowledge and insight, and is facilitated by the use of appropriate CAD-tools. From inspection of root loci or Bode plots, it can be concluded that an integral controller with negative gain will stabilize the loop. After mapping the rate limitation into a bound on the bandwidth as above (step 1), it can be seen that with this controller the maximal possible bandwidth can be achieved, and the gain is then increased until the rate limitation is reached. The results of the two approaches are compared in Fig. 1. Most control engineers will probably agree that the second approach did not result in an inferior performance compared to the first one, but obtained this with a simpler controller, and, with some experience, less design effort. In both cases, one would like the controlled variable to converge faster to its steady-state value. In the classical approach, it is clear what to do: insert an additional lag compensator. It turns out however, that this inevitably produces more overshoot. So here the method reaches its limitations: it cannot be decided without extensive trials whether a more complex controller would allow one to obtain both small overshoot and rapid convergence. (In fact, it can be shown that there is an inevitable trade-off under the given bandwidth limitation, see Appendix 1.) But it is also not obvious how to proceed further using the mixed-sensitivity minimization approach, because the shape of the transient response is not
directly related to the choice of the weighting functions.
r.~/I
t.2
*
Log
.......................................
1
O.II
IH,o mixed sensitivity optimization
I
0.4
O.= 0
Fig. 1. Step responses obtained by IH.o-mixed sensitivity minimization and by classical design techniques This contribution describes a novel approach to controller design which uses IH®-optimization but is more closely related to practical design goals than the standard approach because the desired transient behaviour enters directly into the cost function. The goal is to integrate the benefits of IH**-optimization with the easy control of the transient behavior provided by classical design techniques. 2. D E S I G N S P E C I F I C A T I O N S In this section a minimal set of specifications is stated which are encountered in any realistic design problem, and these specifications are mapped into frequency-domain constraints. This is the basis for the derivation of a modified IH®-optimization problem which encompasses this set of specifications in a transparent way. Consider the standard feedback configuration shown in Fig. 2, where V(s) = _~s)W(s) +_S(s)_Z(s). (1)
W(s=) ~
Z(s)
Fig. 2. Standard control loop The starting point here is the observation that in almost all practical cases, there is a hierarchy of design specifications: 1. Hard constraints due to plant uncertainty, measurement noise, and actuator limitations. The
Design of Robust Control Systems
simplest representation of these constraints is a norm bound on the reference-to-output or complementary sensitivity transfer function or transfer matrix: ]T(jco)
(2a)
367
is equivalent to finding the minimal value of v for which the frequency response SOco) lies, for each value of co, in a disk with radius r(m) and center c(m). c(co) and r(r~) are related to Wl(jm) and W2Om)
by W2(J°~) 2
or
VT(jm)_~jo~)WT(j~ ) < I
(5a)
w,(jm)2+ w (jm) 2
(2b)
where H is the maximal singular value. r(~)= c(~
One well-known special case of (2a) is
W2.jto. 2 +c(co)-I
.
(5b)
T(jco) <12(m ) , (Doyle and Stein, 1981) where 1m is a bound on the multiplicative plant uncertainty. (2a,b) usually implies a specification of high-frequency roll-off. 2. Feedback control is always introduced to counteract unmeasured disturbances and plant uncertainties. This means small values of the sensitivity transfer function or transfer matrix over some set of frequencies and not too large values elsewhere, hence
S(jm) < as(m)
(3a)
or
So the centers of the circles move from 0 to I on the real axis as ]W I [decreases and [W2[ increases, ff [W I [>>[ W2 [, (4) is approximately equivalent to
s(j )
(3b)
(3a,b) in particular implies a certain bandwidth of the control loops and the asymptotic behavior as o~ --, O. 3. In general, control systems are judged upon by testing time responses, particularly step responses. Good damping, reasonable peak values and fast convergence to the steady-state value are general quality criteria. The step response is determined, for the most part, by the behavior of the control loops near the gain crossover frequencies. As the previous example shows, reasonable specifications of Ir(jco) l at high frequencies and of I S(jco) I at low frequencies alone do not ~ t c c a good transient response. As far as the reaction to reference steps is concerned, the desired behaviour can always be achieved by introducing a shaping filter in front of the loop, i.e. by using a 2-degree-of-freedom structure. The reaction to disturbances, e.g. load changes, however is still determined by the loop transfer functions S and T. How are these specifications reflected in mixed sensitivity minimization? It can be shown that the minimization of v in
v = sup{ WI(jm)S(j~) 2 + W:(jm)T(jm) : }
(4)
(6a)
and ff IW: I >>1Wl I, (4) is approximately equivalent to T(j~) <,T(jro)= ~r~. W:(jc0) -1.
(6b)
I>>lwel
Usually, IW 1 will hold as co --} 0 to antee good disturbance rejection whereas [>>[Wl[will hold as co -} ~ to ensure robustness against unmodelled or unknown dynamics. So it is relatively easy to represent the specifications 1 and 2 in the form (4). [
_Vs(jm)_S(j~)Ws (jco) <1.
w,(jm)-',
t
t
By minimizing (4), the weighted fit of S0~) and a real reference function which increases from 0 to 1 is optimized. This implies that there can be little control of the phase of the closed-loop system because S(j~) can never be close to such a real function for all frequencies. The weights influence both the reference function (the centers of the circles) and the size of the circles in a not too transparent manner. If standard simple weighting functions are used, the behaviour in the intermediate frequency range and thus the step response is not well controlled, because the resulting circles there are large. This argument holds as well in the multivariable case where not only the transient behaviour of the main loops but also the coupling terms may be of interest. In order to achieve better control of the transient behaviour near gain crossover the open-loop frequency response should be close to that of a reference model which provides the desired closedloop step response. From this, the third specification T(jco)- M(j~) < ~M(~)
(7a)
in the scalar case o r VM(j~)[_T(j~)- M(j~)]WM(jm) [ < 1 in the multivariable case results.
(7o)
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S. Engell (9)
In (7a,b), the specification of the transient behaviour is in terms of T(jCO) which is the response to a reference step in the 1-dof structure. In a 2-dof structure, 1-T defines the response to disturbances at the output, Gp . ( l - T ) the response to disturbances at the input of the plant. A model for T is used here because the relation of the transfer function to the step response is well known from the classical design rules and their extensions (Engell, 1991). For a scalar plant which is minimum-phase except for a pure delay and one dominating unstable pole or one dominating rhp zero, a second-order system is usually sufficient as a reference model because additional requirements can be incorporated into (2) and (3). Second-order models can easily be parametrized from time-domain or time and frequency domain specifications. In the multivariable case, the choice of M is more difficult. Often, one will specify a diagonal transfer matrix M using scalar techniques for the specification of the diagonal elements, and diagonal weighting matrices to account for the different bandwidths. By the three specifications (2), (3), (7), all basic design requirements are represented in a a uniform manner, ff the specifications (2) and (3) are fixed from the practical problem, the designer will try to optimize the transient behavior within these constraints, e.g. by choosing M(s) such that the bandwidth roughly agrees with the constraints imposed by (2) and (3) and minimizing ~ . Alternatively, the effect of smaller values of es at low frequencies on the attainable fit to the reference model can be investigated. In the SISO case, the set of specifications (2a), (3a), (7a), constitutes a 3-disk-problem which is not analytically tractable. It may also not have a solution at all, ff the three disks do not overlap at each frequency, i.e. the specifications are not compatible. However, it frequently suffices to meet the specifications in different frequency ranges. (2) will be imposed at high frequencies, (3) at low frequencies, and (7) in the intermediate range. So the design problem in the scalar case is formulated as: Find a stabilizing controller such that
S(jCO)-c(CO) < r(CO) with
VCO
fi
c(CO)= -M(jCO)
ICOl COb
Ob
ICOl---COz and
I
(10)
ICOl-< COb
s(CO)
r(CO)= M(CO)
COb< ICOl< CO,"
/
{F--'T(CO)
ICOl CO,
Note that c(CO) is complex. The problem of the compatibility of the specifications is then reduced to the compatibility of (la) and (7a) at COband (3a) and (7a) at COz.In general, one should choose COb,COzsuch that M(jCOb) = ~S(COb),
M(jCOz) = ~;T(COz)
(11)
in order to have a smooth transition from one specification to the other. In the multivariable case, a similar representation with a piecewise defined center function matrix and right and left weighting matrices is V(jCO)[_C(jCO)-_S(jCO)]W(jCO)[< 1
(12)
3. T R A N S F O R M A T I O N I N T O A N I H o o - O P T I M I Z A T I O N PROBLEM The existence of a stabilizing controller which satisfies (8) or (11) can checked to any desired precision by the introduction of rational approximations to c(CO)and r(CO)or V, C, and W. In the scalar case, the approximation is specified as follows: outside the intervals ~ s _ < ~_o___< 1 COb ~ S
o_~_<
'
~T~
1 ~T
'
c(CO) and r(CO) are approximated with prescribed maximal approximation error -
inside these intervals, the approximation error is bounded and at most as large as ~ / 2 times the difference of the two functions which are pasted together at COb or COz.
(8)
So the approximation introduces significant (but still bounded) errors only in small intervals around COb, coz, and arbitrarily small errors elsewhere. The relative values of the errors outside the transition ranges and the size of the transition range are related for a given order of the approximation, but both can
Design of Robust Control Systems
be made as small as desired by increasing the order of the approximation. Suitable approximations can then be constructed using the so-called Zolotarev-functions (Piloty, 1954). K(s/%) denotes the n-th order Zolotarevfunction with comer frequency col. It satisfies, among all functions of the same order, [K(jco')t < $
for ~ 1 < ~
(13a)
IK(jo')l>8-'
for ~']>l/~f~
(13b)
for the maximal possible value of I] for given 8 or, vice versa, the minimal value of 8 for given [~. The functions, first obtained by Zolotarev have the form -jz ]2I i=1 1+ b~z2
(n odd)
"= l+b2z2
(18)
where Es(s), E~s), Er(s ) are rational functions with modulus es(CO), eM(CO), ~(0)) on the jco-axis, and { }+ denotes the stable minimum phase transfer function having the same modulus on the jo-axis. After Vc(s) and Vr(S) are chosen, it can be checked whether
plls(Jo)- vo(jo)l-qv,(Jo)l} o
(19)
~0
(14a)
(n even)
(14b)
(z=co/%). The parameters b i depend on ~ or 8 (Caner, 1940; Pilot),, 1954). K(z) increases monotonically in the transition interval and has modulus 1 at z = 1. From these functions, rational approximations to the unit-step function on the j~axis (s' = s / % ) are constructed:
(_i)vs,kKn(/~ ]
This problem can be solved using any of the available techniques, e.g. Pick-interpolation (Pick, 1916; Engell, 1988), polynomial methods (Meinsma and Kwakernaak, 1991) or state-space algorithms. The minimal value of ,c can in fact be computed without iteration. The minimization of r in (19) is identical with the solution of the "generalized I ~ optimal control problem" which was introduced independently by Grimble (1989) because of computational advantages but not discussed with respect to the choice of weighting functions.
(15)
P(')= 1+(_l)~,(s,k)k Kn(~) Apart from the factors s'k these are the well-known elliptichigh-pass filterfunctions.They satisfy P(jco')
Vr(s) = (Es(s) [1 - Pb(~b/ + EM(S)Pb(~-b)[1 - Pz(~z/] +
can be satisfied for ~c < 1 with a stabilizing controller. Vr is stable, but Vc in general is not.
(n-1-'----~)_(b ~ + Z2)
='
369
for k o ' l < ~
(16a)
I - P ( j o ' ) <~'1 k. 8 forlco'l>--1/f~. 1-8
(16b)
If k and n are both even or both odd, T is given by the condition that P must remain finite on the imaginary axis, and P(jco) is real. Otherwise, 7 can be chosen freely, and depending on this choice the Nyquist plot of P(jco)is a semicircle from (0, j0) to (I,j0) in the upper or in the lower half-plane.
4. DESIGN E X A M P L E The use of this IH®-optimization approach with 3 independent design goals is illustrated for the example discussed in Section 1. The hard constraint in this case is given by the actuator rate limitation. It can be mapped into the frequency domain specification
T oI=bI ,C o) The next step is to specify a model for the transient behaviour. Starting as simply as possible, the fact that the plant has an unstable pole and unstable zeros was ignored at first and a standard second order model with 10% overshoot was chosen and the bandwidth was roughly adapted to (20). The result was
The factors s'k are necessary to prescribe the convergence of the error for co - , 0 and co - , ~. The rational approximations V¢(s), Vr(S) finally are
Vc(s)=
Pb S 1 M ( S ) 1 (~bb) f-
Pz s
(~z~]
+Pz s
(~zz) (17)
MI(s ) -
1
1+ 23s + 400s 2
Now if under the constraint (20), a controller can be found which gives a system which is a good approximation problem would be solved. This can be using ~s,(O) = 1- M,(jo)
(21) stabilizing closed-loop of M 1, our checked by (22)
370
S. Engell
below rob and minimizing aM (a constant value was used here in general) The result was, however, that the convergence of the step response to the steadystate value is unsatisfactory (as before). So S0ro) must be made smaller at low frequencies. To achieve this,
rr,'k
1.2
1.0
0.1
0.1,
Es2(s) _ ks(S+COb)
(23) 0.4
S2
with k s such that used.
.......
¢$1
-.-*-.-
u=+ • 0.00,~1
--.o--.
u= • o . o o e J+ ¢sz
Hi
Es2(J%)= 1-Mr(job)was
0.2
0
was again minimized for different values of %. roz was chosen as 0.04 which is the frequency where ~r(O~) is equal to 1. The orders of the functions Pb, Pz were 7 and 8 with ~/~= 0.7. The resulting frequency responses for minimal values of eM are shown in Fig. 3, the corresponding step responses in Fig. 4. It turns out that the desired behavior cannot be matched for any reasonable value of %. This is due to the unstable pole of the plant which was not included as a zero in M v One may pick a value of rob as a reasonable compromise, e.g. rob= 0.004, and then simplify the resulting controller. The allowed disks and the optimal solution for this value are shown in fig. 5 as an illustration of the geometry of the 3-disk specification and the well-known behavior of the solution to he always on the edge of the specified region.
'
,~0,
'
,~0,
'
,,=,'
'
,+'=.
'
,o=,
Fig. 4. Step responses corresponding to Fig. 3
2
S(ja))
MI,~s2
Vc(jm) 1
+
0.5
~ P:.<:" . ..... :~ ,, : (/", ,/,," '-, ,,", ' / / : . "2-22; --.,'.'.. i ! ,i' ~/li -3 ,. \ ", ","-'-=";',','-/"/'
/4 // I/,?'
-0.5 The orders of the approximations must be relatively large to provide a rapid transition from one specification to the next. Otherwise, there is significant interaction and the goal to evaluate three independent specifications cannot be met. The exponents k must be such that the approximation error remains small compared to 6 s and %. The order of K(z) must at least be 3, but 5 is preferable. In Fig. 6, the moduli of Vc and Vr are shown together with those of S and 1 - M 1 for gs2 with rob = 0.004 as in Fig. 5. More details on the effect of the approximation can be found in (Engell and Oukhai, 1991).
"~1
-0.5
o25
d
i
125
2
Fig. 5. Prescribed regions and optimal frequency response for rob = 0.004
2O 0 -2O -40 -60:
IStjmll.lmlj=ll
d8 lO
. . . . . . . .
i
. . . . . . . .
i
-8(
. . . . . . . .
i
. . . . . . . .
j, `I'` 0J I+ ' ' `
1~
. . . . .
ISljul
.I(X •
O.
•
o
~
__._ -120 -140
-lO,
-160
-2o. "1 ~ 0 d 4
-3o,
-40.
-50.
-50
/
//
//
---,-.-
..... IVctju)l .......o.... IVr (ju)l
i
--x---
1
........
, 10.=
,
, ,,,,,
I1 - 141 ( j u } l
. . . . . . 104
,. . . . . . . . . . . 10.1
foal=
I
tOa
Fig. 6. V c and V r for Mr(s), ¢ob = 0.004
...... o;.ooo,r' ,
,//4:/ . . . .
w= = 0 . 0 0 4 ]
,' .......... "",/
'-",
•, g l / . , , t
it)";I
i
i
i i iiii
I
i0"I
. . . . . . . .
i
1'0"I
. . . . . . . .
rod/=
100
Fig. 3. [ S(jm)l for M 1 (s) and Ss) or SS2 f o r v a r i o u s values of rob, ~M minimized
To investigate more deeply whether a better performance is possible, the next step is to respect the unstable pole in the reference model by choosing it such that M(Pu)=l. The simplest reference model
Design of Robust Conti"ol Systems which achieves this is a second-order transfer function with a zero in the nominator. It was specified such that the gain at % is 1 to be compatible with the bandwidth limitation (20), and the rise-time was optimized for a desired overshoot of 10%.
2
S(j~)
-
1.5
371
-.
M2
vc(jco)
1
0.5 Using this reference model instead of M~ however gave unsatisfactory results. The resulting value of is so large that there is no resemblance of the step responses. This is due to the fact that above ~z, ST is much smaller than the gain of the reference model and therefore its phase cannot be matched below o~z because of the higher required roll-off above coz. Therefore the reference model was modified by adding an additional pole at 0.06 to obtain roughly the same roll-off as ~ . In order to get 10% overshoot, M 2(s) 1+ 116s (24) 1+ 127s+ 3524s 2 + 2 8 0 6 0 s 3 was chosen. Note that M2(Pu) does not have exactly the desired value at p,, because this would result in too much overshoot again and the goal was to explore whether this is inevitable or not. No additional attenuation at low frequencies was specified. Then the Optimal closed-loop system which minimizes the error between T and M.2 in the intermediate frequency range and satisfies the stability conditions has the step response shown in Fig. 7. The approximation of M 2 is very good (compare Figs. 8, 9). If more disturbance attenuation at low frequencies is asked for, the error between T and M 2 increases again. Essentially, the results then are the same as obtained with M r
0 '~" \"~.",'";'-:'.." ,:,/i!/ ..... ,,~..."............. ~..:,;y ,~,~:........... :.~,,,
-0.5 -I
-1.5
-2i
-1
6
i
2
Fig. 8. S(j~) and prescribed regions for minimal Using the proposed approach, a clear picture of the inevitable trade-offs was obtained for this example independent of a particular structure of the controller. The problem encountered using t h e classical frequency domain design technique was found not to be a result of that particular technique but a consequence of the plant dynamics, i. e. the unstable pole, and the specification (20). This can also be proven analytically (see Appendix 1). The specification of three independent performance measures allowed us to explore the limitations on the attainable performance in a precise manner. The example also shows that it may be difficult to get as good results as those obtained by purely interactive design by using standard optimization methods.
d8 20 0 .20 4O -6O 410
0. .12G
0.6
--x---
11 - M 2 ( j w } I
-140 / .ll
0.~
0.2
1o-,
I~
I~
d-,
Io~
1o,
I~ ,.¢ko,
Fig. 9.]Vc and]Vr forM2(s),G M minimized s~O. lO00m 1200.
14oo,
I~,
I~LOO. )~O(Om
Fig. 7. Step response obtained for M2(s)
All controllers can be reduced to order 2 using the technique described in (Engell, 1991) without loss of performance.
5. S U M M A R Y A novel formulation of the controller design problem in the framework of IH®-optimization was described. This formulation includes all three major design specifications: robustness, disturbance rejection, and
S. EngeH
372
transient response. The procedure can be extended to the multivariable case in a straight-forward manner by pasting the piecewise definied weighting matrices or center function matrices together as in the scalar case. However, to prevent an explosion of the order of the controller, the specifications (2b), (3b), and (7b) should be pasted together at the same frequencies for all elements of the matrices.
APPENDIX 1: Attainable step responses for unstable SISO plants Assume a SISO plant has an unstable pole at s = p in the rhp. Then for internal stability of the closed-loop system shown in Fig. 2, T(p) = 1
(m)
~gT(t)e-Ptdt = 1 0
where gT(t) is the impulse response of T. Integration by parts yields (A3)
0
As
~e-ptdt = p-l, 0 co
f[1-hT(t)] e-ptdt = 0.
Poles
Zeros
+.0011 -.0125 -.0727+/-j.0778 -1.044+/-j.610
-.0023 -.0129 -.207 -1.23 +6.14 +/-j 5.96
DC-gain: 41.49 Rate limitation: du/dt < .00222 for unit reference step
Acknowledgement: The author is very grateful to I. Postlethwaite and S.D. O'Young for the details of the example.
(A1)
is necessary. Let hT(t) be the step response for reference steps of the closed-loop system. Assume hT(0) = 0 (strictly proper plant) and hT(t) --> 1 for t oo (steady-state accuracy). From (A1),
~hT(t)e-Ptdt = p-I
APPENDIX 2: Data of the example (plant model)
(A4)
0
Formula (A4) restricts the attainable step responses: the integral of the control error, weighted by exp(pt) must vanish. As hT(t) is below 1 for small values of t, this area must be compensated by values above 1 for larger t, and this compensation is discounted as t -~ oo, so the step response cannot show an infinitesimally small error for a long time to satisfy (A4). If an unstable pole and actuator limitations are both present, the initial error integral of the step response is bounded from below, and hence there must be a positive control error the time integral of which is at least as large as the initial error integral, hence either large overshoot and rapid convergence to the steady-state value or small overshoot and slow convergence occur, as encoutered in the example.
REFERENCES Cauer, W. (1940). Bemerkung fiber eine Extremalaufgabe von E. Zolotareff. Zeitschr. f. Ang. Math. u. Mech., 20, 358. Doyle, J., and Stein, G. (1981). Multivariable Feedback Design: Concepts for a ClassicalModern Synsthesis. IEEE Tr. on Automatic Control 26, 4-16. Engell, S. (1988). Optimale lineare Regelung. Springer-Verlag, Berlin. Engell, S. (1991). Towards a Complete Frequency Domain Design Methodology. 1st IFAC Symposium on Design Methods of Control Systems, Preprints, 66-72, Ztirich. Engell, S., and Kerker, R. (1989). Controller Design by Approximation of Ideal Compensators. Proc.1989 American Control Conference, 18991900, Pittsburgh. Engell, S., and Oukhai, A. (1991). Model Matching Under Hard Constraints: A Case Study. Proc. 1st European Control Conference, 709-714, Grenoble. Grimble, M.J. (1989). Generalized IH~Multivariable Controllers. IEE Proceedings D, Vol. 136, 285-297. Meinsma, G., and Kwakernaak, H. (1991). IH~oOptimal Control and Behaviours. Proc. 1st European Control Conference, 1741-1746, Grenoble. Pick, G. (1916). Woer die Beschr~inkung analytischer Funktionen. Math. Ann., 77, 7-23. Piloty, H. (1954). Zolotareffsche rationale Funktionen. Zeitschr. f. Ang. Math. u. Mech., 34, 175-189. Postlethwaite, I., O'Young, S.D., Gu, G.-W., and Hope, J. (1987). IH~-Control Systems Design: a Critical Assessment. IFAC World Congress, Prepr. Vol. 8, 328-333, Miinchen.