- Email: [email protected]
Time-Optimal Control and Disturbance Compensation for a Class of Hybrid Systems
3c-0l 2
Copyright © 1996 IF AC 13th Triennial World Congress. San Francis(O. USA
TIME-OPTIMAL CONTROL AND DISTURBANCE COMPENSATION FOR A CLASS OF HYBRJD SYSTEMS
Steran Pellersson, Bengt Lennartson Cmuml Engineering Lllb, Chalnrers Universify of Technology .1'-412 96 GotIJel1burg, Sweden e-mail: sp.bl(dJcolllrol.chalmers.se
Abstract: In this paper the design of a time-optimal controller [or a class o[ hybrid systems where the continuous part is modelled by integrator processes is outlined. The controller may be used for dassical tracking problems where the goal is to follow a given trajectory. However, since the control law can be expressed ar.; a function of the present continuous state it can also compensate for disturbances, acting as a traditional feedback controller where the continuous state is forced hlwards a desired set point. Keywords: Time-optimal control design, Hybrid system, Integrator process.
I. INTRODUCTION
In many systems, changes in the structure result in discontinuities in the dynamics. The changes may e.g. be caused by discrete actuators or hy inherent process discontinuities. Systems that include a combination of continuous behavior and such discrete features are called hybrid systems. There have been several propositions for the modelling of hyhrid systems. Common for these frameworks arc that the continuous evolution is affected hy a discrete event system (automata) which results in changes in thc continuous dynamics. Onc modelling approach is to use an interface that couples the discrete event system to the separated continuous part. An alternative approach is phase-transition structures developed by computer scientists. The purpose of this paper is to design a time-optimal controller for a specific class of hybrid systems. Antsaklis et al. (1993) have developed a method for control law synthesis based on the interface model, and an algorithm for direct generation of control events based on the hyhrid automata model
has been developed by Tiltus and Egardt (1994). This second approach is limited to the class of integrator processes.
In this paper the design of a time-optimal controller based directly on the hybrid model and the continuous and discrete specilkations is outlined. The controller may be used for classical tracking problems where the goal is to follow a given trajectory. However, since the control law can be expressed as a function of the prescnt continuous statc it can also compensate fordisturbancm-;, acting as a traditional feedback controller where the continuous state is forced towards a desired set point. The method is based on a geometric approach and is restricted to integrator processes. The paper is organized as follows. In Se<.:tion 2 the model to be used I'm the controller design is c1assil1ed and specific continuous and discrete features arc discussed. The design procedure is explained in section 3. Finally, aspects concerning an introduced switch strateg:y arc discussed in Section 4. An extended version of this paper can be found in (Petlcrsson and Lcnnarlson 1995a).
4706
2. MODEL CLASSIFICATION This section discusses which type of systems come into question for the time-optimal controller design.
2. J. Continuolls be/wvior
Ch)
(a)
(c)
For a hybrid system, it is complicated in general to solve optimal control pmblems. Ilowevcr, for the special case of integrator proc.:esses. time-optimal solutions can be generated as will be shown.
Fig. I. Ca) Tank process. (b) Real automaton. (c) Equivalent automaton if every intermediate state can be passed arhitrarily [&..,t.
For an integrator process the stale evolution can be expressed hy the differential c4uation
Vn. The equation that describe the evolution of the process
.r(l) = .,,(0) =
!Vi
"'0.
+ v(l),
I,. S t '''(If) - xf
< 'Hl
is given hy the integrator process
(I )
whcrCl~ E !JlII,ll.lj EI1 = {WII'U12"",Wm}. Thevcctur 11(t) is an unknown disturhance allccting the system. In general, the initial state :T(J is not known beforehand. However, the lina} state :1: f is a given adjustable set point which may be (I fUllction of time. It is assumed that the state :r is possible to measure directly.
The continuous specific.ation is introdu<.:cd in the form of state constraints for the continuous variables. This can formally be defined hy k relations, /),(,,) S 0, where each Pi : 3(11 .-)0 ~, i = 1, ... , k. All state (onstraints together, p(:r.) S 0, I' : ~1! ----+ ~J~~', form a set l{ which is called the allowable region. lh guarantee thc.t a solution exists for an arbitrary initial state :l'f) E 1(, the allowahle region must satisfy certain properties. For instance, if the allowable region is convex or. more generally, star-shaped (sec (Peltcrsson and Lennartson L995a) for definition), solutions will exist. This topic is more thnnHlghly diS(.'ussed later on.
2.2. Discrete behavior The set !l represents the discrete part or the hybrid system and consists of 111 control vectors. The discrete part can be modelled by a discrete automaton where each control vector represents a discrete state. For the design procedure to work, no restrictions on the switch se<)uence between the different discrete states can he introduced, i.e. it must be pO&"iihlc to switch from every control vector to every other. This is a reasonabLe assumption when the discrete part consists of independent actuators, for instance on/off valves. This is illustrated hy the following example. Example I Consider the tank process given in Fig. la. There are two different input flows. ti,l and g/J, to the tank, each controlled by a two-state actuator (openfclosed). The slates of the system are chosen as the parlial volumes l!"A and
] . .:> [ 11.',,:
= 'Wj,
where the control vectors W j arc given by the four different control vcchlr combinalions~ namely
111\
= [ :; ] IV, = [
q~
] 1113 = [
q~J
]1114
= [
~~
]
(2) The cnrrespondingautomaton is shown in Fig. Ib where the discrete states represent each of the control vectors. As can be seen there arc no connections between the diagonal states since it is assumed that both valves cannot be switched at the same time. Howevcr~ if we assume that it is possible 10 pass an intermediatc state arbitrarily fast the diagonals should also be connected. This is shown in Hg. le. • The m control vectors in the Sl't n represent those control vectors that arc allowable and are therefore called Ihe allowable controll'eclOrs. These control vectors are given by the discrete specification and other p(}.ssifJ/e cOIlIml vectors arc regarded as forbidden slales. The forbidden states can be considered as limitations in the discrete dynamical behavior. The reason for cxc.::1uding some of the possible control vectors may have a physical background related to the state of the actuators.
3. TIME OPTIMAL CONTROLLER A"ume an integrator system given by (I) where x E !Rn, E U = {WI' 11/1, .. . ,11)11/}. n is the set of allowahle control vedors and there arc no restrictions on the switch scquence for the m control vectors in that set. Givcn a sct point ;c f the task is to design a controller that transfers the system from an arhitrary point ;1:0 to the set point :r:f, in minimal lime. The designed controller can he used both for classical tracking prohlems and for disturbance compensation. The tracking prohlem is obtained if the set point ;r f varies with
'/Vj
4707
time. However, if:./' f is fixed, at least picccwise, and we strive to stay at ;f f despite the disturbances, the c:ontrollcr is used as a regulator. Further on, we will show that these different cases can be regarded in the same way. If the initial state is known beforehand it is possible to generate the time-optimal control law by solving a linear pn)gramruing problem and introducing a swilch strategy (PeUersson and Lcnnarlson I 995h). However, the control law must be updated when the actual trajectory differs too much from the calculated because of disturbances. To solve this problem a time-optimal control law will be formulated that is a function of the present state. The approach is to to create a state space partitioning where different regions uniquely define which control vectors arc to be used to reach the set point :rf in minimal lime without violating the state constraints. Many of the used mathematical notations and concepts can he found in hooks all topology anti geometry e.g. (Preparata and Shamos 1985).
In (his section, the characterization or the solution of the inlegralor process is given. Al Ihis poinl, Ihe dislUrbance v(t) is nol included. Inlegraling (I) wilhoul 1'(1), the sel of points reachable from :1"0 up to time t is expressed by
Ru (:ro ,
n = {:ro + L ;=1
Theorem 1 Let "'[ # To and ass!lme that 3t > 0 "!lch that "[ E RnCro'!). Tiren ":1 E 01!,,(:ro, t') where I: denotes
tile minimal lime to ,.each ;r f.
It!
TiUli; LTi
:S
f, T1
~ 0,
Wi
E U}
i-I
(3) Each Tj ? 0 represents the tutal ame the corresponding control vector Wi E U is active. (xn, l) is denoted the reachable sul,set Jhml :ro up In lime t. At time t = n, Rn (:1"0,0) = {:ro} and as lime increases Ru(xo, t) grows. 'lh characterize Ru(:ro,t), note that :/'0 may be moved outside the set R11 (:rO, t) and if the time-scale also is normalized, Rn (:ro, f) can he wriUcn as
no
1I"(:r,,. I)
expands continuously with time t hut preserves its geometrical shape, cr. Fig. 2 which indicates reachable subsets Jor the process in Example I. The dolled lines indicale the directions of the control vectors given by (2). The directions are from l() and outwards.
For a fixed lime f., the following theorem characterizes points in Ru (;ro, I) that arc reached in time t but not in less time. The proof of the theorem can he found in (Pcttcrsson and I.ennartson 19950).
3.1. Characterizatioll of the ~oIUli()fl
III
fig. 2. Illustration of how Ru (xo, t) expands continuously.
= "0 + 11,,(0, I) = :r" + IR,,(O, 1)
(4)
The set R,,(O, 1j, will be called the normalized reachable subset and it is a convex combination set of the vecturs (points) 101, '/If2, ... ,'tl ' /!!. The sets Rn (0,1) and Ru(:r.o, t) are polyhedrals, sce (Preparala and Shamos 1985). Given two polyhedrals Iln(:r", t, j and N,,(:,o, t,) where fl is a non-emply scl, il 1(,IIows from (4) that I[ < 12 implies Ihal R"(:r,,,I,) c 11,,(:<0,1 2 ), This is most easily seen if wc change the system of coordinates and place the origin at "'". Then Ihe two scls IIn(O, I,) and IIn((l, /2) arc relaled as R,,(O,ld = hUlI(O,I,j, implying Ihal every point in R" (n, Id is continuously scaled by Ihe time t, It,, compared to the set Ro(O, (d. Thus, when Lime t increases, the polyhedral Ru(O, f). and thus R11 (:rO. f) in the olt! coordinates,
In view of Theorem J, it is of interest to characterize the control vectors which should be used to reach a point x f on the boundary, OR[)(:!"o, I). Therefore, change the coordinale system and place the origin at :[:0. Let P be the sel n scaled by time I, i.e. P = tn = it.,::r Ell}. Now Ihe polyhedral R,,(O, t) may be described by the convex hull of Ihe points in 1', Co(1'). Each face of the polyhedral R,,(O, t) is Ihe convex hull of a suhset of points in P. Dcnotc F C P as the largest sct of points descrihing an arbitrary face. Further let S = {x E j' Cl': x ~ F} (i.l·. S is an arbilrary subsel of P where the points F are not included). The control vectors that should be uscd 10 reach a poinl.TI on OR"(:"o,t) arc now given hy the following theorem and the proof can be found in (Pcltcrssnn and LennartsonI995a). Theorem 2 A.mlme Ilrat x [ E i!R" (0, t) is on tire face described by the po;",... ill F. Tlren X! E Co(F). FUI'llrer
,,:[
~
CotS).
It should he noted that there is not in general a unique timcoptimal solution for the control prohlem. The reason is that all points in OR" (:ro, t) can be reached by vector addilion (ef. (3)), wilh individual veclOr Ienglhs proportional to the 10tal time each control vector is act ive. Since vector addition is
4708
it
commutative npcration. i.e, itllocs not maller in which order we add the vectors. the lack of uniqueness folluws. There is a unique lime-optimal solution if and only if xI is a vertex of the polyhedral Ru(x", t) . This corresponds to the case when unly onc of thc VCl'tors in U is used 10 obtain (he timeclptimal solution.
if we merely recopy the poi nts fonning the vertices of the convex hull. Further, since nur main goal is lu notain the time-nptimal state space partitioning, (he explicitly knowledge of which specific vcctors that are not time-optimal is of minor importancc. This implies that we are only concerned wilh Problem 2.
The lack of uni4ucness is important since because of this, the controller design problem may be divided into thrce stcps. Firstly, the state space is partitioned into regions in which different lime-optimal control vectors are used. In this step no attention is drawn to the state constraints. Secondly. if certain conditions arc given for the stale constraints. it is shown that the sel point :rf can he reached without violating these . Finally, a switch strategy is introduced for the selection of a specific order in which the vectors should hc used.
The convex hull problem in two ancJ three dimensions has a classic geoOletnl: interpretation . Most of the algorithms constructed so far are given for two dimensions only. In the higher dimensional cases the geometric objects arc not as simple as their two dimensional counterparl.... so one must seck a careful organization or the computation of the faccls of the convex hull in orc.ler to cut the likely overhead. There are algorithms for higher dimensions. However, describing these goes hcyond the scope of this papcr, we refer to e.g. (Prep. rata and Shamos 19H'i) for more details.
SCUJOlI
3.2. c,elJCrmiml o/Ihe fime-CllJlimal COIlIIVI t'eClOrs Since the time-uptinml points are on the boundary of the reachable subset anti tllis scl expands continuously, Ihe same VCl:tors in it are used to generate Ihe points on a specific race or th e houndary of the reachable suhset at arbitrary limes, Thus, considering the normalized reachable suhsct Ru(O , 1), the prohlern is to search those control vectors that give a lime-optimal solution. Further, given the time-optimal control vectors. it is necessary to obtain a description of ORn (U.l) In know which vectors arc to he llsed to reach a specilic point nn the houndary_ Therefore the second problem is to construct the convex hull, Co(U), which givcs a l'ompletc description of the boundary of the nnrmali:lell rcachaole sunset. Thus wc have the following two problems:
Problem 1 Given a set n of fit points (vectors) in tify th,,,e thal are vertices of C,,{!I). Problem 2 (Jiven a set 11 of fit I')oinls (vectors) in struct its convex hull .
~n,
~II,
3.3. S(
A").'!iumc nuw that a full description of the convex hull of Ihe normalizecJ reachahle suhset has been obtained, Since (4) holds, wc have also a full description of Ihe convex hull of RIl(X", t) hyjust scaling the obtained coordinates with time t and transforming it hy :1:0. Since the goal is to reach the Hnal state :r f' either if it is varying or lixed. from an arhitrary initial state :rll. wC now start at x f at t = 0 and look backwards ill time to see how the rcachahle subset increases. Thus for a fixed time -/., all points"n 01/,,(0, -t) will reach :r: J in time f.. For a particular initial slate Xo tim e '. away from :"/, the vectors F to he used arc Ihose for which:
idencon-
Prohlem L TIleallS tn identify a suhset of U which consists of t;me-of1limal c:omrol vectors. This set is denoted by !l",,/. ThusDR,,((),l) = Dfl"",., (0. I). Theremainingvectors! if any, arc vectors in either the interior of Un(O, L), luf.(Ru(O, 1)). or for houndary vectors, they can be written as a convex comhination of vectors in ~ ~(lf!I' Using any of the vectors in lnf.(Ru{O, 1)) results in a control law that is not lime-optim:ll .
e
The above problems are well known in the diS{;iplin e known as computatioflal geomel1y. The first is denoted th e etlreme point prohlem anulhe second th e t"OIll'cr /rull prohlcm (Prcparala and Shamos 1985)_ The solutiun (If these problems is tightly couplellto tJata structures and algorithms, 'Ih find the extreme points and convex hull we note that I'roblem ., is asymptotically al IC
Since the reachable subsct expands continuously, with respect to time -I, wc naturally obtain the region partitioning by dcsc:ribing the n-dimcnsional t'tlllCS origin at :r./ and going through the points describing the faces of the normalized reachahle suhset (wilh respect to Ihe system of coordinales at x f), where the sign of the vectors in nO ,11is changed. Thus, if the current state is in one of these cones, the vectors to he llsed for the lime-optimal control arc given by those which define the corresponding cone. To illustrate the ahovc procedure we return to Example 1 and consider Fig. J. To obtain the state spaee partitioning. a point strictly between -'//':I and -tlJ1 reach ", in optimal time by using the vectors 10:1 and 1/)4. Th erefore the whole region given by the conc with origin at VI and gning through the points -W:I and '-W,h reach V, ill optimal time by lIsing the vCClorsw3 and W4 (cr. Region' in Fig. 3",). In the s.1 me way wc obtain the region where till and 1tJ4 arc used (cf. Region 2 in Fig. 3h). If the actual slate i ~ a point that coincides with any of the dolted lines, there is a unique solutiul1 where the corresponding vector is used,
4709
and Lennartson 1995a).
RU
Theorem 3 Let K E ~1I he a cio,\'ed star-shaped set with x f belonging to tIle kel'fJel of g, Assume that a poilll :r." E [)f( has beefl reached where the gradient of J( is defined at :fp' Let the lime-optimal ,mimiml be given hy
-2w" ;r:, =
Ca)
"
LTiWi
+ xp
j:o::l
Then there always exists at least olle cOflll'01 vector Wi whiclr has all allowable direction.
W2 "
IRegion
A vectorwi that has an allowable direction means that if xp E then 3, > 0 such thal "r + 'W; E K. Nole that if xr E [nt(K) all directions are allowed direclions.
f(
l!egiOri
1I
Ch) Fig. 3. (a) /lo(li, -I). (b) Slale space partilioning into re-
gions. from the ahove. wc can conclude that the tracking and regulator problem can be regarded in the same way since if l/f varies we only adjust the calculatt~d regions by the current value of
l~f'
3.4. Conditiolls for tile slale ("O/l.'itraims At this point, the stalc space has heen partilioned into regions where difTcrcnl control vectors arc uscUln reach the set point ;(1 in minimal lime, This procedure takes no concern of the given state constraints. In this section, we will introduce the stale constraints and give conditions for lhcse to make it possible to still reach the set point :r f with the obtained stalc space partitioning.
The slale constraints together form a set It,.', which is the allowable region, It is easy to construct an example which shows that in some points on the houndary DK it is possihie to get «stuck" (i.c. it is not possible to reach the scl point "f wilhoul passing oulside Ihe allowahlc region), cL for instance (PeHcrsson and Lcnnartson I995h). Points on the bOllnd
From the theorem we can conclude that the set point :r f must helong to the kernel of ];,." to guarantee that a solution exists for all points in l\" which are interior points of K or boundary points where the gradient is defined. Consequenlly, if ;t f moves outside the kernel of K. but is still in [(, there are regions in the state space for which we cannot guarantee a solution to exist.
4. SWITCH STRATEGY When the tlilTerent regions are calculated, we know which of the time-optimal control vectors in {lopt are to be used in each region. However, the spccilic order of using them must be intrm.luccd by a switch strategy. There are many aspects to consider when specifying the order of the control vectors.
4.1. Switch .wrategv when 110 diswrballces affect lire system If no disturbances affect the system, some aspects to consider when designing a switch strategy are: • to keep down the numher or switches;
• to avoid points on the bounllary of the allowable region, OH, where it is possible to get stuck. The firsl item can be physically mOlivaled by rcmemhcring that every switch is a result of a state change for some of the discrete actuators and thus, 10 minimize wear, they should change slate as seldom as possible. It can be shown that at most n of the f time-optimal control vectors in no]'l, are needed to reach the Hila I slate :1; f (Pettcrsson and Lennartsoll 1995h), If no stale omstraints are considered the system then needs to change control vectors at most (n - 1) times. Howcver, if state constraillts arc considered, these will
4710
perhaps increase the number of times the control vectors arc changed. The best solution would be to Hnd a switch strategy such that no constraints at all are reached, since then at most (n - 1) switches arc necessary. Generally, it is not easy to minimize the number of switches, and the difficulty increases with (he order of the system. 1I0wever, this problem may be solved by dynamic programming. The second item is motivated by the ohservation in Section 3.4 that there are points on DI\' where it is possible to get stuck, i.e. it is impossihle to reach the set point x f without violating any of the state constraints. Thus, the introduced switch strategy must prevent Ihese points heing reached.
4.2. S)'\-'itd, slrateg}' wire" dislUrhallce.~ alTec' the system If disturbances affect the system it is still important to keep down the numher of switches, as mentioned previously. I-Iowevc-r, there are also some further aspects to consider: • to avoid outer surfaces (if any) and lJK;
5. CONCLlIS[ONS
A time-optimal controller design procedure has been presented. The systems under consideration arc integrator processes with a discrete input signal represenled by a set of control vectors. The design procedure can be divided into three steps. Firstly, the state-space is partitioned into regions where different time-optimal control vectors arc used. Secondly, if the additional state constraints form a star-shaped set with the desirell sel point helonging to its kernel, it has heen shown that the obtaincd stale space partitioning can still be used to reach the set point without exceeding the allowahle region. Finally, since there is in general no unique solution for a specilk initial state, a switch strategy is introduced. We have discussed some aspects that may be important for a suitable selection of thhis strategy. I [owcver, this subject will be investigated further on, hoth theoretically and practically, in future work. Acknowledgements: Valuable discussions with Prof. Ho Egardt are very much appreciated. This work has been financially supported by the Swedish Research OlUncil for Engineering Sciences (TFR) under the project number 92-185 .
• to avoid chattering/unstable strategies.
The outer surfaces in the lirst item arc houndaries to ("'r. -I) VI > O. Note that there are no outer surfaces iLrf E illf(R o"" (:rf, -I)) for an arbitrary small t. [I' the trajectory is close to an outer surface or aK, the distur~ ham::es may cause the trajectory to pass outside the reachable suhsct or the allowahlc region. Thus, these surfaces must if Ilossihle he avoided. Note that since the whole houndary of [{ should he avoided, this also includes points where it is possihle to get stuck.
uR"",.,
The next item. to avoid ehatteringlunstahle strategies, may happen if the strategy in different regions is designed to steer toward the same surface. The intention is to use the control vectors that spans this surface to reach the set point ;r:/. However, hccause of the disturhance the trajectory will not stay in this suh-space. Instead an approximated hehaviour of the trajectory on the surface occur (sliding mode) with the result that the control vectors often are changed (chattcring). Chattering strategies are not desirable since they wear out the actuators. But even worse is that the system may hecome unstahle. The chattering phenomenon and the in~ stability problem may he solved by introducing a hysteresis around the surface where chattering may occur (Pcttcrsson amJ Lennarlsnn 1995a).
REFERENCES Anl"klis, P.l., JA Stiver and M. Lemmon (1993).llybrid system modeling and 3utOl1Omous t:ontrol systems. In: Jlybrid Systems (R.L. Gro"man, A. Nerode, A.P. Ravn and 11. Rischcl. Eds.). pp. :1(,(,-392. Lec!. N. in Olmp. Sci. 736. Springer VcrJag. Pettersson, S. and 13. Lennartson (I 995a). Time-optimal controt and disturhance compensation for a class of hybrid systems. Technical Report [-95/010. Control Engineering L
Note that this section only points out certain aspects that should be considered when to introduce the switch strategy. The specific application JinalJy decides how the acluaJ strategy is implemented.
4711
Copyright © 1996 IF AC 13th Triennial World Congress. San Francis(O. USA
TIME-OPTIMAL CONTROL AND DISTURBANCE COMPENSATION FOR A CLASS OF HYBRJD SYSTEMS
Steran Pellersson, Bengt Lennartson Cmuml Engineering Lllb, Chalnrers Universify of Technology .1'-412 96 GotIJel1burg, Sweden e-mail: sp.bl(dJcolllrol.chalmers.se
Abstract: In this paper the design of a time-optimal controller [or a class o[ hybrid systems where the continuous part is modelled by integrator processes is outlined. The controller may be used for dassical tracking problems where the goal is to follow a given trajectory. However, since the control law can be expressed ar.; a function of the present continuous state it can also compensate for disturbances, acting as a traditional feedback controller where the continuous state is forced hlwards a desired set point. Keywords: Time-optimal control design, Hybrid system, Integrator process.
I. INTRODUCTION
In many systems, changes in the structure result in discontinuities in the dynamics. The changes may e.g. be caused by discrete actuators or hy inherent process discontinuities. Systems that include a combination of continuous behavior and such discrete features are called hybrid systems. There have been several propositions for the modelling of hyhrid systems. Common for these frameworks arc that the continuous evolution is affected hy a discrete event system (automata) which results in changes in thc continuous dynamics. Onc modelling approach is to use an interface that couples the discrete event system to the separated continuous part. An alternative approach is phase-transition structures developed by computer scientists. The purpose of this paper is to design a time-optimal controller for a specific class of hybrid systems. Antsaklis et al. (1993) have developed a method for control law synthesis based on the interface model, and an algorithm for direct generation of control events based on the hyhrid automata model
has been developed by Tiltus and Egardt (1994). This second approach is limited to the class of integrator processes.
In this paper the design of a time-optimal controller based directly on the hybrid model and the continuous and discrete specilkations is outlined. The controller may be used for classical tracking problems where the goal is to follow a given trajectory. However, since the control law can be expressed as a function of the prescnt continuous statc it can also compensate fordisturbancm-;, acting as a traditional feedback controller where the continuous state is forced towards a desired set point. The method is based on a geometric approach and is restricted to integrator processes. The paper is organized as follows. In Se<.:tion 2 the model to be used I'm the controller design is c1assil1ed and specific continuous and discrete features arc discussed. The design procedure is explained in section 3. Finally, aspects concerning an introduced switch strateg:y arc discussed in Section 4. An extended version of this paper can be found in (Petlcrsson and Lcnnarlson 1995a).
4706
2. MODEL CLASSIFICATION This section discusses which type of systems come into question for the time-optimal controller design.
2. J. Continuolls be/wvior
Ch)
(a)
(c)
For a hybrid system, it is complicated in general to solve optimal control pmblems. Ilowevcr, for the special case of integrator proc.:esses. time-optimal solutions can be generated as will be shown.
Fig. I. Ca) Tank process. (b) Real automaton. (c) Equivalent automaton if every intermediate state can be passed arhitrarily [&..,t.
For an integrator process the stale evolution can be expressed hy the differential c4uation
Vn. The equation that describe the evolution of the process
.r(l) = .,,(0) =
!Vi
"'0.
+ v(l),
I,. S t '''(If) - xf
< 'Hl
is given hy the integrator process
(I )
whcrCl~ E !JlII,ll.lj EI1 = {WII'U12"",Wm}. Thevcctur 11(t) is an unknown disturhance allccting the system. In general, the initial state :T(J is not known beforehand. However, the lina} state :1: f is a given adjustable set point which may be (I fUllction of time. It is assumed that the state :r is possible to measure directly.
The continuous specific.ation is introdu<.:cd in the form of state constraints for the continuous variables. This can formally be defined hy k relations, /),(,,) S 0, where each Pi : 3(11 .-)0 ~, i = 1, ... , k. All state (onstraints together, p(:r.) S 0, I' : ~1! ----+ ~J~~', form a set l{ which is called the allowable region. lh guarantee thc.t a solution exists for an arbitrary initial state :l'f) E 1(, the allowahle region must satisfy certain properties. For instance, if the allowable region is convex or. more generally, star-shaped (sec (Peltcrsson and Lennartson L995a) for definition), solutions will exist. This topic is more thnnHlghly diS(.'ussed later on.
2.2. Discrete behavior The set !l represents the discrete part or the hybrid system and consists of 111 control vectors. The discrete part can be modelled by a discrete automaton where each control vector represents a discrete state. For the design procedure to work, no restrictions on the switch se<)uence between the different discrete states can he introduced, i.e. it must be pO&"iihlc to switch from every control vector to every other. This is a reasonabLe assumption when the discrete part consists of independent actuators, for instance on/off valves. This is illustrated hy the following example. Example I Consider the tank process given in Fig. la. There are two different input flows. ti,l and g/J, to the tank, each controlled by a two-state actuator (openfclosed). The slates of the system are chosen as the parlial volumes l!"A and
] . .:> [ 11.',,:
= 'Wj,
where the control vectors W j arc given by the four different control vcchlr combinalions~ namely
111\
= [ :; ] IV, = [
q~
] 1113 = [
q~J
]1114
= [
~~
]
(2) The cnrrespondingautomaton is shown in Fig. Ib where the discrete states represent each of the control vectors. As can be seen there arc no connections between the diagonal states since it is assumed that both valves cannot be switched at the same time. Howevcr~ if we assume that it is possible 10 pass an intermediatc state arbitrarily fast the diagonals should also be connected. This is shown in Hg. le. • The m control vectors in the Sl't n represent those control vectors that arc allowable and are therefore called Ihe allowable controll'eclOrs. These control vectors are given by the discrete specification and other p(}.ssifJ/e cOIlIml vectors arc regarded as forbidden slales. The forbidden states can be considered as limitations in the discrete dynamical behavior. The reason for cxc.::1uding some of the possible control vectors may have a physical background related to the state of the actuators.
3. TIME OPTIMAL CONTROLLER A"ume an integrator system given by (I) where x E !Rn, E U = {WI' 11/1, .. . ,11)11/}. n is the set of allowahle control vedors and there arc no restrictions on the switch scquence for the m control vectors in that set. Givcn a sct point ;c f the task is to design a controller that transfers the system from an arhitrary point ;1:0 to the set point :r:f, in minimal lime. The designed controller can he used both for classical tracking prohlems and for disturbance compensation. The tracking prohlem is obtained if the set point ;r f varies with
'/Vj
4707
time. However, if:./' f is fixed, at least picccwise, and we strive to stay at ;f f despite the disturbances, the c:ontrollcr is used as a regulator. Further on, we will show that these different cases can be regarded in the same way. If the initial state is known beforehand it is possible to generate the time-optimal control law by solving a linear pn)gramruing problem and introducing a swilch strategy (PeUersson and Lcnnarlson I 995h). However, the control law must be updated when the actual trajectory differs too much from the calculated because of disturbances. To solve this problem a time-optimal control law will be formulated that is a function of the present state. The approach is to to create a state space partitioning where different regions uniquely define which control vectors arc to be used to reach the set point :rf in minimal lime without violating the state constraints. Many of the used mathematical notations and concepts can he found in hooks all topology anti geometry e.g. (Preparata and Shamos 1985).
In (his section, the characterization or the solution of the inlegralor process is given. Al Ihis poinl, Ihe dislUrbance v(t) is nol included. Inlegraling (I) wilhoul 1'(1), the sel of points reachable from :1"0 up to time t is expressed by
Ru (:ro ,
n = {:ro + L ;=1
Theorem 1 Let "'[ # To and ass!lme that 3t > 0 "!lch that "[ E RnCro'!). Tiren ":1 E 01!,,(:ro, t') where I: denotes
tile minimal lime to ,.each ;r f.
It!
TiUli; LTi
:S
f, T1
~ 0,
Wi
E U}
i-I
(3) Each Tj ? 0 represents the tutal ame the corresponding control vector Wi E U is active. (xn, l) is denoted the reachable sul,set Jhml :ro up In lime t. At time t = n, Rn (:1"0,0) = {:ro} and as lime increases Ru(xo, t) grows. 'lh characterize Ru(:ro,t), note that :/'0 may be moved outside the set R11 (:rO, t) and if the time-scale also is normalized, Rn (:ro, f) can he wriUcn as
no
1I"(:r,,. I)
expands continuously with time t hut preserves its geometrical shape, cr. Fig. 2 which indicates reachable subsets Jor the process in Example I. The dolled lines indicale the directions of the control vectors given by (2). The directions are from l() and outwards.
For a fixed lime f., the following theorem characterizes points in Ru (;ro, I) that arc reached in time t but not in less time. The proof of the theorem can he found in (Pcttcrsson and I.ennartson 19950).
3.1. Characterizatioll of the ~oIUli()fl
III
fig. 2. Illustration of how Ru (xo, t) expands continuously.
= "0 + 11,,(0, I) = :r" + IR,,(O, 1)
(4)
The set R,,(O, 1j, will be called the normalized reachable subset and it is a convex combination set of the vecturs (points) 101, '/If2, ... ,'tl ' /!!. The sets Rn (0,1) and Ru(:r.o, t) are polyhedrals, sce (Preparala and Shamos 1985). Given two polyhedrals Iln(:r", t, j and N,,(:,o, t,) where fl is a non-emply scl, il 1(,IIows from (4) that I[ < 12 implies Ihal R"(:r,,,I,) c 11,,(:<0,1 2 ), This is most easily seen if wc change the system of coordinates and place the origin at "'". Then Ihe two scls IIn(O, I,) and IIn((l, /2) arc relaled as R,,(O,ld = hUlI(O,I,j, implying Ihal every point in R" (n, Id is continuously scaled by Ihe time t, It,, compared to the set Ro(O, (d. Thus, when Lime t increases, the polyhedral Ru(O, f). and thus R11 (:rO. f) in the olt! coordinates,
In view of Theorem J, it is of interest to characterize the control vectors which should be used to reach a point x f on the boundary, OR[)(:!"o, I). Therefore, change the coordinale system and place the origin at :[:0. Let P be the sel n scaled by time I, i.e. P = tn = it.,::r Ell}. Now Ihe polyhedral R,,(O, t) may be described by the convex hull of Ihe points in 1', Co(1'). Each face of the polyhedral R,,(O, t) is Ihe convex hull of a suhset of points in P. Dcnotc F C P as the largest sct of points descrihing an arbitrary face. Further let S = {x E j' Cl': x ~ F} (i.l·. S is an arbilrary subsel of P where the points F are not included). The control vectors that should be uscd 10 reach a poinl.TI on OR"(:"o,t) arc now given hy the following theorem and the proof can be found in (Pcltcrssnn and LennartsonI995a). Theorem 2 A.mlme Ilrat x [ E i!R" (0, t) is on tire face described by the po;",... ill F. Tlren X! E Co(F). FUI'llrer
,,:[
~
CotS).
It should he noted that there is not in general a unique timcoptimal solution for the control prohlem. The reason is that all points in OR" (:ro, t) can be reached by vector addilion (ef. (3)), wilh individual veclOr Ienglhs proportional to the 10tal time each control vector is act ive. Since vector addition is
4708
it
commutative npcration. i.e, itllocs not maller in which order we add the vectors. the lack of uniqueness folluws. There is a unique lime-optimal solution if and only if xI is a vertex of the polyhedral Ru(x", t) . This corresponds to the case when unly onc of thc VCl'tors in U is used 10 obtain (he timeclptimal solution.
if we merely recopy the poi nts fonning the vertices of the convex hull. Further, since nur main goal is lu notain the time-nptimal state space partitioning, (he explicitly knowledge of which specific vcctors that are not time-optimal is of minor importancc. This implies that we are only concerned wilh Problem 2.
The lack of uni4ucness is important since because of this, the controller design problem may be divided into thrce stcps. Firstly, the state space is partitioned into regions in which different lime-optimal control vectors are used. In this step no attention is drawn to the state constraints. Secondly. if certain conditions arc given for the stale constraints. it is shown that the sel point :rf can he reached without violating these . Finally, a switch strategy is introduced for the selection of a specific order in which the vectors should hc used.
The convex hull problem in two ancJ three dimensions has a classic geoOletnl: interpretation . Most of the algorithms constructed so far are given for two dimensions only. In the higher dimensional cases the geometric objects arc not as simple as their two dimensional counterparl.... so one must seck a careful organization or the computation of the faccls of the convex hull in orc.ler to cut the likely overhead. There are algorithms for higher dimensions. However, describing these goes hcyond the scope of this papcr, we refer to e.g. (Prep. rata and Shamos 19H'i) for more details.
SCUJOlI
3.2. c,elJCrmiml o/Ihe fime-CllJlimal COIlIIVI t'eClOrs Since the time-uptinml points are on the boundary of the reachable subset anti tllis scl expands continuously, Ihe same VCl:tors in it are used to generate Ihe points on a specific race or th e houndary of the reachable suhset at arbitrary limes, Thus, considering the normalized reachable suhsct Ru(O , 1), the prohlern is to search those control vectors that give a lime-optimal solution. Further, given the time-optimal control vectors. it is necessary to obtain a description of ORn (U.l) In know which vectors arc to he llsed to reach a specilic point nn the houndary_ Therefore the second problem is to construct the convex hull, Co(U), which givcs a l'ompletc description of the boundary of the nnrmali:lell rcachaole sunset. Thus wc have the following two problems:
Problem 1 Given a set n of fit points (vectors) in tify th,,,e thal are vertices of C,,{!I). Problem 2 (Jiven a set 11 of fit I')oinls (vectors) in struct its convex hull .
~n,
~II,
3.3. S(
A").'!iumc nuw that a full description of the convex hull of Ihe normalizecJ reachahle suhset has been obtained, Since (4) holds, wc have also a full description of Ihe convex hull of RIl(X", t) hyjust scaling the obtained coordinates with time t and transforming it hy :1:0. Since the goal is to reach the Hnal state :r f' either if it is varying or lixed. from an arhitrary initial state :rll. wC now start at x f at t = 0 and look backwards ill time to see how the rcachahle subset increases. Thus for a fixed time -/., all points"n 01/,,(0, -t) will reach :r: J in time f.. For a particular initial slate Xo tim e '. away from :"/, the vectors F to he used arc Ihose for which:
idencon-
Prohlem L TIleallS tn identify a suhset of U which consists of t;me-of1limal c:omrol vectors. This set is denoted by !l",,/. ThusDR,,((),l) = Dfl"",., (0. I). Theremainingvectors! if any, arc vectors in either the interior of Un(O, L), luf.(Ru(O, 1)). or for houndary vectors, they can be written as a convex comhination of vectors in ~ ~(lf!I' Using any of the vectors in lnf.(Ru{O, 1)) results in a control law that is not lime-optim:ll .
e
The above problems are well known in the diS{;iplin e known as computatioflal geomel1y. The first is denoted th e etlreme point prohlem anulhe second th e t"OIll'cr /rull prohlcm (Prcparala and Shamos 1985)_ The solutiun (If these problems is tightly couplellto tJata structures and algorithms, 'Ih find the extreme points and convex hull we note that I'roblem ., is asymptotically al IC
Since the reachable subsct expands continuously, with respect to time -I, wc naturally obtain the region partitioning by dcsc:ribing the n-dimcnsional t'tlllCS origin at :r./ and going through the points describing the faces of the normalized reachahle suhset (wilh respect to Ihe system of coordinales at x f), where the sign of the vectors in nO ,11is changed. Thus, if the current state is in one of these cones, the vectors to he llsed for the lime-optimal control arc given by those which define the corresponding cone. To illustrate the ahovc procedure we return to Example 1 and consider Fig. J. To obtain the state spaee partitioning. a point strictly between -'//':I and -tlJ1 reach ", in optimal time by using the vectors 10:1 and 1/)4. Th erefore the whole region given by the conc with origin at VI and gning through the points -W:I and '-W,h reach V, ill optimal time by lIsing the vCClorsw3 and W4 (cr. Region' in Fig. 3",). In the s.1 me way wc obtain the region where till and 1tJ4 arc used (cf. Region 2 in Fig. 3h). If the actual slate i ~ a point that coincides with any of the dolted lines, there is a unique solutiul1 where the corresponding vector is used,
4709
and Lennartson 1995a).
RU
Theorem 3 Let K E ~1I he a cio,\'ed star-shaped set with x f belonging to tIle kel'fJel of g, Assume that a poilll :r." E [)f( has beefl reached where the gradient of J( is defined at :fp' Let the lime-optimal ,mimiml be given hy
-2w" ;r:, =
Ca)
"
LTiWi
+ xp
j:o::l
Then there always exists at least olle cOflll'01 vector Wi whiclr has all allowable direction.
W2 "
IRegion
A vectorwi that has an allowable direction means that if xp E then 3, > 0 such thal "r + 'W; E K. Nole that if xr E [nt(K) all directions are allowed direclions.
f(
l!egiOri
1I
Ch) Fig. 3. (a) /lo(li, -I). (b) Slale space partilioning into re-
gions. from the ahove. wc can conclude that the tracking and regulator problem can be regarded in the same way since if l/f varies we only adjust the calculatt~d regions by the current value of
l~f'
3.4. Conditiolls for tile slale ("O/l.'itraims At this point, the stalc space has heen partilioned into regions where difTcrcnl control vectors arc uscUln reach the set point ;(1 in minimal lime, This procedure takes no concern of the given state constraints. In this section, we will introduce the stale constraints and give conditions for lhcse to make it possible to still reach the set point :r f with the obtained stalc space partitioning.
The slale constraints together form a set It,.', which is the allowable region, It is easy to construct an example which shows that in some points on the houndary DK it is possihie to get «stuck" (i.c. it is not possible to reach the scl point "f wilhoul passing oulside Ihe allowahlc region), cL for instance (PeHcrsson and Lcnnartson I995h). Points on the bOllnd
From the theorem we can conclude that the set point :r f must helong to the kernel of ];,." to guarantee that a solution exists for all points in l\" which are interior points of K or boundary points where the gradient is defined. Consequenlly, if ;t f moves outside the kernel of K. but is still in [(, there are regions in the state space for which we cannot guarantee a solution to exist.
4. SWITCH STRATEGY When the tlilTerent regions are calculated, we know which of the time-optimal control vectors in {lopt are to be used in each region. However, the spccilic order of using them must be intrm.luccd by a switch strategy. There are many aspects to consider when specifying the order of the control vectors.
4.1. Switch .wrategv when 110 diswrballces affect lire system If no disturbances affect the system, some aspects to consider when designing a switch strategy are: • to keep down the numher or switches;
• to avoid points on the bounllary of the allowable region, OH, where it is possible to get stuck. The firsl item can be physically mOlivaled by rcmemhcring that every switch is a result of a state change for some of the discrete actuators and thus, 10 minimize wear, they should change slate as seldom as possible. It can be shown that at most n of the f time-optimal control vectors in no]'l, are needed to reach the Hila I slate :1; f (Pettcrsson and Lennartsoll 1995h), If no stale omstraints are considered the system then needs to change control vectors at most (n - 1) times. Howcver, if state constraillts arc considered, these will
4710
perhaps increase the number of times the control vectors arc changed. The best solution would be to Hnd a switch strategy such that no constraints at all are reached, since then at most (n - 1) switches arc necessary. Generally, it is not easy to minimize the number of switches, and the difficulty increases with (he order of the system. 1I0wever, this problem may be solved by dynamic programming. The second item is motivated by the ohservation in Section 3.4 that there are points on DI\' where it is possible to get stuck, i.e. it is impossihle to reach the set point x f without violating any of the state constraints. Thus, the introduced switch strategy must prevent Ihese points heing reached.
4.2. S)'\-'itd, slrateg}' wire" dislUrhallce.~ alTec' the system If disturbances affect the system it is still important to keep down the numher of switches, as mentioned previously. I-Iowevc-r, there are also some further aspects to consider: • to avoid outer surfaces (if any) and lJK;
5. CONCLlIS[ONS
A time-optimal controller design procedure has been presented. The systems under consideration arc integrator processes with a discrete input signal represenled by a set of control vectors. The design procedure can be divided into three steps. Firstly, the state-space is partitioned into regions where different time-optimal control vectors arc used. Secondly, if the additional state constraints form a star-shaped set with the desirell sel point helonging to its kernel, it has heen shown that the obtaincd stale space partitioning can still be used to reach the set point without exceeding the allowahle region. Finally, since there is in general no unique solution for a specilk initial state, a switch strategy is introduced. We have discussed some aspects that may be important for a suitable selection of thhis strategy. I [owcver, this subject will be investigated further on, hoth theoretically and practically, in future work. Acknowledgements: Valuable discussions with Prof. Ho Egardt are very much appreciated. This work has been financially supported by the Swedish Research OlUncil for Engineering Sciences (TFR) under the project number 92-185 .
• to avoid chattering/unstable strategies.
The outer surfaces in the lirst item arc houndaries to ("'r. -I) VI > O. Note that there are no outer surfaces iLrf E illf(R o"" (:rf, -I)) for an arbitrary small t. [I' the trajectory is close to an outer surface or aK, the distur~ ham::es may cause the trajectory to pass outside the reachable suhsct or the allowahlc region. Thus, these surfaces must if Ilossihle he avoided. Note that since the whole houndary of [{ should he avoided, this also includes points where it is possihle to get stuck.
uR"",.,
The next item. to avoid ehatteringlunstahle strategies, may happen if the strategy in different regions is designed to steer toward the same surface. The intention is to use the control vectors that spans this surface to reach the set point ;r:/. However, hccause of the disturhance the trajectory will not stay in this suh-space. Instead an approximated hehaviour of the trajectory on the surface occur (sliding mode) with the result that the control vectors often are changed (chattcring). Chattering strategies are not desirable since they wear out the actuators. But even worse is that the system may hecome unstahle. The chattering phenomenon and the in~ stability problem may he solved by introducing a hysteresis around the surface where chattering may occur (Pcttcrsson amJ Lennarlsnn 1995a).
REFERENCES Anl"klis, P.l., JA Stiver and M. Lemmon (1993).llybrid system modeling and 3utOl1Omous t:ontrol systems. In: Jlybrid Systems (R.L. Gro"man, A. Nerode, A.P. Ravn and 11. Rischcl. Eds.). pp. :1(,(,-392. Lec!. N. in Olmp. Sci. 736. Springer VcrJag. Pettersson, S. and 13. Lennartson (I 995a). Time-optimal controt and disturhance compensation for a class of hybrid systems. Technical Report [-95/010. Control Engineering L
Note that this section only points out certain aspects that should be considered when to introduce the switch strategy. The specific application JinalJy decides how the acluaJ strategy is implemented.
4711