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Time-optimal feedback system controlling a discontinuous dynamic object reaching a moving target
Automatica.Vol.30, No. 12. pp. 1937-1941,1994
Pergamon
ElsevierScienceLtd Printed in Great Britain 0005-1098/94$7.00+ 0.00
0005-1098(94) E0033-E
Brief Paper
Time-optimal Feedback System Controlling a Discontinuous Dynamic Object Reaching a Moving Target* WLADYSLAW
HEJMOt
Key
Words--Automatic control; closed-loop systems; optimal control; sliding states; variable structure systems.
where x is a position, y is a velocity of the mechanism, f is a function of motion resistances, u is a measurable control function, K is a positive constant. In order to describe the largest possible class of motion resistances, in particular all types of friction, we assume f ( y ) to be piecewise continuous. Moreover, lul -< 1 (technical constraint), Ill -< K - e, 0 < e < K (condition of system (3) to be controllable). The system (3) describes the dynamics of a broad class of engineering systems (Hejmo, 1987). The discontinuity of the right-hand side of (3) makes it impossible to apply the classical theory of optimization under a minimum time criterion. Hejmo and Kloch (1984) have solved this problem using the theory of differential inequalities. The method of regular synthesis applied to the solution leads to the following time-optimal closed4oo p system transferring any initial state Zo=(xo, Yo)e R2 to any final one (target) zl = (xl, Yl) ~ N2 (Hejmo and Kloch, 1984, 1989):
Abstract--The paper deals with a feedback system created by the use of the method of regular synthesis of an optimal control system which should bring a non-linear and discontinuous planar object to a moving target in minimum time. In real, practical circumstances perturbations such as unexpected and undefined motion resistance variations, parameter changes and errors in measurements of states impinge on this system, which may cause the created time-optimal closed-loop system to generate limit-cycles and be stopped in a state other than the required final one. It is shown that while using the so-called ~:-solution of the discontinuous differential equation, the synthesized closedloop system may become insensitive towards the above perturbations. Some suggestions are made as to practical applications.
1. Introduction GIVEN A CONTROLLED dynamic system:
~=f(z,u),
ZeN",
u E U c O ~ m,
the regular synthesis (Brunovsky, 1974) of control structure for such a system leads control function u : R " ~ U satisfying properties. (a) Each time-optimal solution of (1) solution of the closed-loop system (CLS)
.~=y,
(1)
x(0)=x0;
f(y)+K, = f(y)-K,
a time-optimal to a feedback the following
(x,y) E T + U R +, y(0)=yo, (x,y) E T - t 3 R - , y(O)=yo,
(4)
where T +={(x,y):x=h(y),yyl}, R+={(x,y):x h(y), y E R1}, h is of class C(R1), h(y~)= xb and
is a standard
the derivative of h satisfies almost everywhere (a.e.):
=f(z, v(z)).
(2) (b) Each standard solution of (2) is a time-optimal solution of (1). Obviously, (2) describes the dynamic behaviour of a CLS displacing (1) from any initial state Zo to any target state z~. To justify the construction of such a controller, the following reasons are generally given: (i) there is no need to compute the optimal control for every new initial state separately; and (ii) the controller acting upon (2) is sensitive to instantaneous perturbations, i.e. if at any instant of the process the system is deviated from its optimal trajectory, the remaining portion of the process will again lead to the desired final state (target) and will be optimal with respect to this new initial state. In this paper we will work with the following nonlinear and discontinuous differential equation: 2 = y , x(0)=x0; ) = f ( y ) + K u , y(0)=y0, (3)
(a) h'(y)=F÷(y)
a.e. on
(-%Yl];
(b) h ' ( y ) = F ( y )
a.e. on
[Yl, 0¢),
where h'(y) =
~yY)
Y
, F+(y) f ( y ) + K ' F_(y)
(5) Y
f(y)-K"
The switching curve T = T + U T U z, and the regions R +, R - play here the same role as in the classical case of an object described by: x" = f ( ~ ) + Ku, f ~ Cl(N1).
Remark 1.1. From each Zo E R
[or R ÷] there starts an optimal trajectory which runs over R - [or R ÷] and then reaches T + [or T-] in finite time. From the point of intersection there starts the time-optimal trajectory which, lying totally on T + [or T-], reaches z~ in finite time. • Practically, there exists a motionless target z~ = (x~,yl), x~ E R t, Yl = 0 and a moving one, zl = (xl, Yl), xl E R I, Yl ~ 0 which is shown in Fig. 1. The paper by Hejmo and Kloch (1989) deals with the dynamic behaviour of the CLS (4) reaching the motionless target by external perturbations acting upon the controlled object (3). This paper also deals with the CLS (4), but is concerned with the case of a moving target and operating in the presence of real perturbations. It is shown that in practical applications such a structure may generate limit-cycles and moreover is able to stop continuously the controlled object in a point located out of the target. The main goal of this work is therefore to obtain a rule of synthesis for such a feedback
* Received 20 July 1991; revised 6 July 1993; revised 9 November 1993; received in final form 8 February 1994. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor T. Ba~ar under the direction of Editor Huibert Kwakernaak. Corresponding author Wladyslaw Hejmo. Tel. +48 12 33 03 00; Fax +48 12 33 84 51; E-mail [email protected]. i Cracow University of Technology, Institute of Control Engineering, ul. Warszawska 24, 31-155 Crakow, Poland. 1937
1938
Brief Papers
T-
T\
z
Y
©
© ©
©
'
TI~ ~
T
+
FIG. 1. Time-optimal switching curve.
system, which would avoid the difficulties arising from a direct application of the earlier theory to the moving target case.
2. Preliminaries The right-hand side of the CLS (4) is discontinuous in terms of its argument. In practical applications however, for technical reasons equation (5) cannot be performed, and the standard solutions of (4) thus become inappropriate (Brunovsky, 1974). Namely, they cannot characterize all the motions that can occur in the system modelled by (4) and, conversely, not all standard solutions admit physical interpretation. For discontinuous differential equations, the most complete definition of solution is that of Filippov (1964). In this paper both the standard and Filippov class of solutions will be used.
Definition 2.1. Let x : l ~ R n (1 is an interval in R ~) be an absolutely continuous function on each compact subinterval of I. Then x is called: (a) a standard (or Caratheodory or ~) solution of a differential equation
£c(t) =g(t,x(t)),
g : ~ l ×R'---~R",
(6)
iff x satisfies (6) almost everywhere on 1; (b) a Filippov (or ~ ) solution of (6) iff k(t) e OZ(g(t,x(t))) almost everywhere on I, where operator F is given by the formula:
OZ(g(t,x)) = n
n
~->0 ~ ( z ) = O
cvxg(t,(x +eB)\Z).
B is the open unit ball in R", I~(Z) is the Lebesgue measure of the set Z, cvxM denotes closure of the convex hull of M ~ I~" (Hajek, 1979). • Both ~ and ~-solutions of the CLS (4) admit the same physical interpretation. Setting in (3) u = 1 and u = - 1 we get the following systems playing an essential role in what follows:
Lemma Z2. Given the system (7) [or (8)], from each z0 there starts a unique R-solution of (7) [or (8)] q+(t;zo) [or q (t;Zo)], t E [0, 0e). The proof results directly from Theorems 4.1 and 5.3 in the work by Brunovsky (1974). • Remark 2.3. (Properties of the solutions q+ and q .) (a) Let y 0 < 0 . Then there exists a finite time 0 < 4 such that y+(t, yo)O on (4, oo). y+(t, yo) is increasing on [0,~), but x+(t, Zo) is decreasing on [0, 4] and increasing on [4, ~)- If Yo -> 0 then x+(., Zo) and y+(., Y0) hold the same properties as above in the interval [4, oo). (b) Let y 0 > 0 . Then there exists a finite time 0 < t l such that y_(t, yo)>O on [0,4), y ( 4 , y o ) = 0 , y (t, y o ) < 0 on (4, ~). Furthermore, y (t, yo) is decreasing on [0, ~), but x (t, z0) is increasing on [0,4] and decreasing on [4, ~). If yo-<0 then as in (a) x+(.,zo) and Y+(.,Yo) hold the same properties as defined above in the interval [4, ~). •
Remark 2.4. From (7), (8) and Lemma 2.2 it arises that in the state plane the trajectories of the solutions q+ and q are defined by the functions x =g+(y) and x = g (y), respectively, resulting from the Lebesgue integrals of the following differential equations: dg~(y)_ (a) ~ F+(y)
a.e. on
[Y0, 2);
( b ) ~ d g - ( Y ) - ~ ( v ) .-
a.e. on
(-~,Yo]-
(9)
Remark 2.5. Lemma 2.2 together with (4) implies that there exists a time 0 < t ' < zc such that from each zo e R + [or R ] there starts a unique R-solution of the system (4), q + ( t ; z 0 ) ~ R + [ o r q ( t ; Z o ) ~ R ], t ~ [ 0 , t'). •
3. Dynamic behaviour of the feedback system
Notation. (a) Any solution of the CLS (4) starting from Zo ~ R 2 will be denoted by q(t;Zo) = (x(t, Zo),y(t, Yo)). (b)
Basic features of the dynamics of CLS operating over (4) have been considered in the paper by Hejmo (1987). Here, attention will be focused on some singular phenomena that can appear in the solutions of (4). Results of theoretical analysis interpreted from a practical point of view lead to the formulation of some suggestions for technical applications.
Solutions of the systems (7) and (8) will be denoted by q+(t; z0) = (x+(t, Zo), y÷(t, Xo)) and q (t; Zo) = (x_(t, Zo), y-(t, yo)), respectively. (c) If Yl > 0 then we define T +, by T+= T ~ U T ~ where T~-={(x,y):x=h(y),y<-O}, T~= { ( x , y ) : x = h ( y ) , y ~ ( O , yl)}. But, if y , < 0 then we define T - , by T = T { U T ~ where T{={(x,y):x=h(y),y->O}, T ~ = { ( x , y ) : x = h ( y ) , y E ( y , , O ) } (Fig. 1). (d) A point intersection of the trajectory T + or T - with the x-axis will be denoted by z' = (x', 0) = (h(0), 0). •
Theorem 3.1. For a given time-optimal CLS (4) and (5), let Yl > 0 [or <0]. Then we have: (i) There exists a finite time 0 < 4 such that from each Zo E T2~ [or ET£] there starts a time-optimal, but non-unique R-solution q+(t; Zo) e T~ [or q (t: z(~) E T 2 ]. t E [ 0 , q ) , q + ( t t ; Z o ) = Z l [ o r q (tt;z~)=zl]. (ii) There exists a finite time 0 < t 2 such that from each zoe T~- [or E T2] there start non-unique R-solutions
k=y,
x(0)=xo;
f=f(y)+K,
y(0)=yo,
(7)
~=y,
x(0)=Xo;
~,=f(y)-K,
y(O)=yo.
(8)
1939
Brief Papers q (t;Zo)•R [or q + ( t ; Z o ) • R + ] , t • [ 0 , t2), q _ ( t a ; Z o ) • T~ [or q+(t2; Zo) • Ti-], which are not time-optimal ones.
are able to find a technical interpretation of the dynamics of the CLS (4) operating in real circumstances.
Proof. (i) The proof of this part results immediately from (4), (5a), (9a) [or (4), (5b), (9b)], Lemma 2.2 and Remark 2.3. (ii) From Lemma 2.2 and Remark 2.3, it follows that there exists a finite time t2 > 0 such that q (t; Zo) • R - [or q+(t;Zo)•R+], t • ( 0 , t2). But (4) implies too, that its q-solution q (t;Zo) • R - [or q÷(t; Zo) • R÷], t • (0, t2). Extending this solution to the boundary of a domain where it exists we get: q ( t ; Z o ) ~ R [or q + ( t ; z 0 ) • R ÷ ] , t • [ 0 , tz), q (t2;zo) • T ? [or q+(t2;Zo) • Ti-]. Optimal solution contained in (i) and non-optimal one in (ii) is obvious. •
Remark 4.1. If inaccuracy ( l l ) occurs, then from any initial state Zo E R2\zl there starts the trajectory of the unique q-solution which spirals around the target, zl and the point z' = (h(0), 0). This trajectory may tend either to a limit cycle (with z~ and z' in its interior) or may encircle the points z~ and z' indefinitely many times, while diverging. •
Practical identification of the real motion resistances function f ( y ) meets with difficulties. Consequently, the model of the object taken for CLS theoretical synthesis is an inaccurate mapping of this object. Hence, because of inexact identification of the function f(y), the switching curve generated by a real CLS (4) may become 'more' or 'less' steep than the time-optimal one. The above cases of the switching curve inaccuracy are defined as follows: (a) h ' ( y ) < F _ ( y )
a.e. on
[Yl, ~);
(b) h'(y)
a.c. on
(-:¢,Yl];
(a) h ' ( y ) > E ( y )
a.e. on
[yj, o~);
(b) h'(y)>F÷(y)
a.e. on
(-~,Yl].
(10) (11)
Theorem 3.2. Given CLS (4), let inaccuracy (11) hold and Yl > 0 [or <0]. Then we have: (i) There exists a finite time 0 < t l such that from each Zoe T~ [or • T ~ ] there starts a non-unique q-solution q+(t;Zo) • R ÷ [or q_(t;Zo) • R-], t • [0, tl), q+(tl;Zo) • T [or q_(fl; Zo) • T +] which is not a time-optimal one. (ii) There exists a finite time 0 < t2 such that from each Zo• T ] [or •T~-] there starts a non-unique %solution q (t;z0) • R - [or q+(t;Zo) • R+], t E [0, t2), q-(t2;Zo) ~ T~ [or q÷(t2; Zo) • Ti-] which is also not time-optimal. Proof. The proof of (i) results immediately from (4), ( l l b ) , (5a) [or (4), ( l l a ) , (5b)], Lemma 2.2 and Remark 2.3. (ii) can be proved in the same way as was done in the proof of T2 in Theorem 3.1. • Theorem 3.3. Given CLS (4), let inaccuracy (10) be valid and Yl > 0 [or <0]. Then we have: (i) From each zo • T [or • T +] there starts the unique ~-solution, the trajectory of which lies totally on T - [or T +] and reaches the target zl in finite non-minimal time. Moreover, none of the q-solutions start from Zo • T - [or • T ÷] for increasing time. (ii) From each Zo • T~-\z' [or •T~\z'] there starts the unique 5~-solution, the trajectory of which lies totally on T~[or T1] and reaches z' = (h(0), 0) in finite non-minimal time. Moreover, none of the %solutions start from Zo • T { \ z ' [or • T?\z'] for increasing time. (iii) From z' there starts a trajectory of the unique ~-solution q(t;zo)~z', t • [0, ~) and none of the qsolutions start from z' (stop of movement). (iv) There exists a finite non-minimal time 0 < t~ such that from each z~'~• T~ [or • T~] there starts a trajectory of the unique q-solution q (t;74)) • R [or q+(t;Zo) • R+], t • (0, tl) and q-(h;Zo) • T~ [or q+(tl;zo) • Ti-]. Proof. The proofs of (i), (ii) and (iii) follow the same logical pattern as that used in the proof of Theorem 3.5 in the paper by Hejmo (1987). The proof of (iv) parallels that of (ii) in Theorem 3.1. • 4. Mismatch problems The real time-optimal controller forms its own switching curve which may differ considerably from that required, e.g. due to computational errors and unknown variations in motion resistances. This switching curve inaccuracy together with errors in state identification will be called 'mismatch problems'. These may cause intolerable behaviour for the CLS (4). Referring some results of the paper by Hejmo (1987) to phenomena shown in Theoems 3.1, 3.2 and 3.3, we
Remark 4.2. Technical interpretation of the ~-solution will be done for Yl > 0 only, the case Yl < 0 being analogous. If the CLS (4) operates with inaccuracy (10) then from each z6 • T - there starts a trajectory of the unique ~-solution. On the other hand, (4) implies that from each z6 • T - there starts the q-solution q_ the trajectory of which should penetrate into R + and implies also that none of q exists in R +. Practically, the trajectory of this solution on leaving z6 • T penetrates into R* where it is immediately forced to repenetrate T (Fig. 2(A)). The real CLS (4) and (10) generates a trajectory that starts to oscillate around T - with a certain frequency and amplitude depending on the delay-time inherent in the switching operation, which evidently exists in every real structure. This trajectory of ~--solution is therefore a limit of the real oscillatory process (sliding, chattering) when the delay time tends to zero, i.e. when the frequency tends to infinity. The same interpretation may be offered for ~-solutions starting both from z6 • T~Xz' (Fig. 2(B)) and z'. The ~-solution therefore becomes a generalized description of the sliding process which has an essential practical meaning. The ~-solution is independent of whatever variation of F_(y) and F+(y) if only (10) is fulfilled. This evident fact will be used in what follows for practical switching curve formation. • Analysis similar to that done in the paper by Hejmo (1987) leads to"
Remark 4.3. Given CLS (4), (10) by y l > 0 , let us define subsidiary sets: T , = {(x, y) :x = g_(y), y >-Yl}; M - = {(x,y):h(y)-yl}; T,={(x,y):x=g+(y), y-<0}; M +={(x,y):g+(y)
,.._A ~./"
~y /
"-L~'-.~
q+
÷
z1
© X
©
B
FIG. 2. Directions of emission of solutions.
1940
Brief Papers Trn
x
~.
~Y
©
t C
x T+, "~.
ZI3
M÷
FIG. 3. Sliding switching curve.
x" < x' = h(0)] in finite time (Fig. 3). The control process for the case y~ < 0 is analogous. • Remark 4.4. Mathematical phenomenon for the occurrence of non-unique %solutions will be interpreted for y~ > 0 only, the case y~ < 0 being analogous. In the real CLS, because of technical reasons (Hejmo, 1987) the states which belong exactly to the switching curve T~ may be identified by the control system as belonging to the region R - . So, the CLS (4) attributes the control function u = - 1 to these states, that generates the ~-solution q the trajectory of which penetrates into R - and afterwards intersects the switching curve Ti ~ in the point ~ • T~\z'. If the switching curve satisfies (5) [or (10)] then from z7 there starts the %solution [or ~-solution] which transfers the state in finite time along T~ to z'. (a) If (5) occurs, then from z' there starts the ~-solution, the trajectory of which transfers the state along T~ and may bring it in finite time to the same point Zo • T2~ in which the movement of the CLS began. If Zo • T2~ is identified, with even the least systematical error, as belonging to the set R , then the CLS (4) generates the limit-cycle a limit-curve of which contains z' in its interior and does not contain z~. Thus, this CLS becomes unable to reach the target z~ (Fig. 4). (b) If inaccuracy (10) occurs then z' becomes an 'end point' of the movement. The CLS (4) is stopped there, because the unique if-solution remains in z' in an infinite time interval. •
Ti FIG. 5. Errors in state measurement.
may recognize the state z~ = (x~., y~) ~ T as belonging to T. Thus, in zE the switching operation is executed. Let the CLS (4) identify the co-ordinates of the state with an error p > 0, to > 0 such that (x~ -4-p, y~ ± to) • T. We will consider a case y ~ > 0 only. Let T and T~ satisfy (10) but TJ do (11). (a) From each z,. such that ( x ~ . - p , y , - t o ) • T and ( x , + 0, Y~ + to) • T - there start, respectively, the trajectories Tm and Tp of the unique ~-solutions which are not able to reach z~. These trajectories can reach only the states forming a certain neighbourhood of zt. The smallest one is a ball B(zl, e) where radius e = (p2 + to2)u~ Trajectories T,, and Tp reach in finite time the boundary B of the B(z~, e) in the points z ~ = (xl + p , yl + to) • B and Zp = (xl - p , yl - to) • B, respectively (Fig. 5). However, from each z, such that (x~ - p, yE - to) • T/- and (x~ + p, y~. + to) • T~" there start, respectively, the trajectories Ti~,,, and T{t, of the unique .~-solutions that reach in finite time points z., = (x' - p, - t o ) and zt~ = (x' + p, to) identified by the CLS as z' belonging to the x-axis (Fig. 5). (b) From each z,- such that (x~-p,y~-to)•T~ and ( x ~ + p , y ~ + t o ) • T ~ there start, respectively, the trajectories T~m and T~p of the unique qg-solutions q+ which start from z " and Zp and reach in finite time z + = ( x l + p , y l - t o ) • / ~ and z ~ = ( x l - p , y ~ + t o ) • / ~
Remark 4.5. The CLS (4) operating in real circumstances
,y - . . Tm
® T2p/1¢//' # ¢#'¢"
T2 l I
/
+ T2m
+
\ FIG. 4. Non-unique solutions.
FIG. 6. Target set and the state trajectories.
Bdef Papers respectively. It is obvious that from each z~ ¢ T ~ there starts the trajectory of the unique C~-solution q+, which runs across R + and intersects T - in a finite time. From z, e T~o there starts the trajectory of the unique qg-solution q_+ reaching in a finite time either T - or B. Which set ( T - or B) is reached by the trajectory of that Cg-solution q+ depends on the values of the state measurement errors p and to (Fig. 6). •
5. Concluding remarks and suggestions for applications In practice, the errors p, to depend on the technical means applied in control structure. Remarks 4.1-4.5 imply the following principles of CLS synthesis. (1) We replace zl by the target set B after estimating maximal values of p, oJ. (2) The shape of T - [or T +] and T~- [or Ti-] must satisfy (10) but T~ [or T~] must satisfy (11) for all possible variations in resistance function
f(Y). For each Zo e R2\B the CLS synthesized in accordance with the above principles ensure the target set reaching along the unique trajectory in finite time independent of variations in motion resistances. The time it takes to reach the target differs from the minimal one, and the difference depends on how much the real switching curve differs from the
1941
time-optimal one. Besides, the control process is as close as possible to the optimal one. Examples in the work by Hejmo and Kloch (1989) show a comparison of the time-optimal process with the suboptimal one generated by CLS operating with an inaccurate switching curve.
References Brunovsky, P. (1974). The closed-loop time-optimal control. SIAM J. Control, 12, 624-634. Filippov, A. F. (1964). Differential equations with discontinuous right-hand side. Trans. Am. Math. Soc. 42, 199-231. Hajek, O. (1979). Discontinuous differential equations. J. Differ. Eqns, 31, 149-185. Hejmo, W. (1987). Stability of a time-optimal closed-loop system with parameter changes. Int. J. Control, 45, 1161-1178. Hejmo, W. and J. Kioch (1984). On the time-optimal problem of positional control with discontinuous resistances of motion. RAlRO-System Analysis Control, 18, 329-341. Hejmo, W. and J. Kloch (1989). On the perturbations in a time-optimal closed-loop system. Ann. Polonici Mathematici. 50, 37-52.
Pergamon
ElsevierScienceLtd Printed in Great Britain 0005-1098/94$7.00+ 0.00
0005-1098(94) E0033-E
Brief Paper
Time-optimal Feedback System Controlling a Discontinuous Dynamic Object Reaching a Moving Target* WLADYSLAW
HEJMOt
Key
Words--Automatic control; closed-loop systems; optimal control; sliding states; variable structure systems.
where x is a position, y is a velocity of the mechanism, f is a function of motion resistances, u is a measurable control function, K is a positive constant. In order to describe the largest possible class of motion resistances, in particular all types of friction, we assume f ( y ) to be piecewise continuous. Moreover, lul -< 1 (technical constraint), Ill -< K - e, 0 < e < K (condition of system (3) to be controllable). The system (3) describes the dynamics of a broad class of engineering systems (Hejmo, 1987). The discontinuity of the right-hand side of (3) makes it impossible to apply the classical theory of optimization under a minimum time criterion. Hejmo and Kloch (1984) have solved this problem using the theory of differential inequalities. The method of regular synthesis applied to the solution leads to the following time-optimal closed4oo p system transferring any initial state Zo=(xo, Yo)e R2 to any final one (target) zl = (xl, Yl) ~ N2 (Hejmo and Kloch, 1984, 1989):
Abstract--The paper deals with a feedback system created by the use of the method of regular synthesis of an optimal control system which should bring a non-linear and discontinuous planar object to a moving target in minimum time. In real, practical circumstances perturbations such as unexpected and undefined motion resistance variations, parameter changes and errors in measurements of states impinge on this system, which may cause the created time-optimal closed-loop system to generate limit-cycles and be stopped in a state other than the required final one. It is shown that while using the so-called ~:-solution of the discontinuous differential equation, the synthesized closedloop system may become insensitive towards the above perturbations. Some suggestions are made as to practical applications.
1. Introduction GIVEN A CONTROLLED dynamic system:
~=f(z,u),
ZeN",
u E U c O ~ m,
the regular synthesis (Brunovsky, 1974) of control structure for such a system leads control function u : R " ~ U satisfying properties. (a) Each time-optimal solution of (1) solution of the closed-loop system (CLS)
.~=y,
(1)
x(0)=x0;
f(y)+K, = f(y)-K,
a time-optimal to a feedback the following
(x,y) E T + U R +, y(0)=yo, (x,y) E T - t 3 R - , y(O)=yo,
(4)
where T +={(x,y):x=h(y),y
is a standard
the derivative of h satisfies almost everywhere (a.e.):
=f(z, v(z)).
(2) (b) Each standard solution of (2) is a time-optimal solution of (1). Obviously, (2) describes the dynamic behaviour of a CLS displacing (1) from any initial state Zo to any target state z~. To justify the construction of such a controller, the following reasons are generally given: (i) there is no need to compute the optimal control for every new initial state separately; and (ii) the controller acting upon (2) is sensitive to instantaneous perturbations, i.e. if at any instant of the process the system is deviated from its optimal trajectory, the remaining portion of the process will again lead to the desired final state (target) and will be optimal with respect to this new initial state. In this paper we will work with the following nonlinear and discontinuous differential equation: 2 = y , x(0)=x0; ) = f ( y ) + K u , y(0)=y0, (3)
(a) h'(y)=F÷(y)
a.e. on
(-%Yl];
(b) h ' ( y ) = F ( y )
a.e. on
[Yl, 0¢),
where h'(y) =
~yY)
Y
, F+(y) f ( y ) + K ' F_(y)
(5) Y
f(y)-K"
The switching curve T = T + U T U z, and the regions R +, R - play here the same role as in the classical case of an object described by: x" = f ( ~ ) + Ku, f ~ Cl(N1).
Remark 1.1. From each Zo E R
[or R ÷] there starts an optimal trajectory which runs over R - [or R ÷] and then reaches T + [or T-] in finite time. From the point of intersection there starts the time-optimal trajectory which, lying totally on T + [or T-], reaches z~ in finite time. • Practically, there exists a motionless target z~ = (x~,yl), x~ E R t, Yl = 0 and a moving one, zl = (xl, Yl), xl E R I, Yl ~ 0 which is shown in Fig. 1. The paper by Hejmo and Kloch (1989) deals with the dynamic behaviour of the CLS (4) reaching the motionless target by external perturbations acting upon the controlled object (3). This paper also deals with the CLS (4), but is concerned with the case of a moving target and operating in the presence of real perturbations. It is shown that in practical applications such a structure may generate limit-cycles and moreover is able to stop continuously the controlled object in a point located out of the target. The main goal of this work is therefore to obtain a rule of synthesis for such a feedback
* Received 20 July 1991; revised 6 July 1993; revised 9 November 1993; received in final form 8 February 1994. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor T. Ba~ar under the direction of Editor Huibert Kwakernaak. Corresponding author Wladyslaw Hejmo. Tel. +48 12 33 03 00; Fax +48 12 33 84 51; E-mail [email protected]. i Cracow University of Technology, Institute of Control Engineering, ul. Warszawska 24, 31-155 Crakow, Poland. 1937
1938
Brief Papers
T-
T\
z
Y
©
© ©
©
'
TI~ ~
T
+
FIG. 1. Time-optimal switching curve.
system, which would avoid the difficulties arising from a direct application of the earlier theory to the moving target case.
2. Preliminaries The right-hand side of the CLS (4) is discontinuous in terms of its argument. In practical applications however, for technical reasons equation (5) cannot be performed, and the standard solutions of (4) thus become inappropriate (Brunovsky, 1974). Namely, they cannot characterize all the motions that can occur in the system modelled by (4) and, conversely, not all standard solutions admit physical interpretation. For discontinuous differential equations, the most complete definition of solution is that of Filippov (1964). In this paper both the standard and Filippov class of solutions will be used.
Definition 2.1. Let x : l ~ R n (1 is an interval in R ~) be an absolutely continuous function on each compact subinterval of I. Then x is called: (a) a standard (or Caratheodory or ~) solution of a differential equation
£c(t) =g(t,x(t)),
g : ~ l ×R'---~R",
(6)
iff x satisfies (6) almost everywhere on 1; (b) a Filippov (or ~ ) solution of (6) iff k(t) e OZ(g(t,x(t))) almost everywhere on I, where operator F is given by the formula:
OZ(g(t,x)) = n
n
~->0 ~ ( z ) = O
cvxg(t,(x +eB)\Z).
B is the open unit ball in R", I~(Z) is the Lebesgue measure of the set Z, cvxM denotes closure of the convex hull of M ~ I~" (Hajek, 1979). • Both ~ and ~-solutions of the CLS (4) admit the same physical interpretation. Setting in (3) u = 1 and u = - 1 we get the following systems playing an essential role in what follows:
Lemma Z2. Given the system (7) [or (8)], from each z0 there starts a unique R-solution of (7) [or (8)] q+(t;zo) [or q (t;Zo)], t E [0, 0e). The proof results directly from Theorems 4.1 and 5.3 in the work by Brunovsky (1974). • Remark 2.3. (Properties of the solutions q+ and q .) (a) Let y 0 < 0 . Then there exists a finite time 0 < 4 such that y+(t, yo)
Remark 2.4. From (7), (8) and Lemma 2.2 it arises that in the state plane the trajectories of the solutions q+ and q are defined by the functions x =g+(y) and x = g (y), respectively, resulting from the Lebesgue integrals of the following differential equations: dg~(y)_ (a) ~ F+(y)
a.e. on
[Y0, 2);
( b ) ~ d g - ( Y ) - ~ ( v ) .-
a.e. on
(-~,Yo]-
(9)
Remark 2.5. Lemma 2.2 together with (4) implies that there exists a time 0 < t ' < zc such that from each zo e R + [or R ] there starts a unique R-solution of the system (4), q + ( t ; z 0 ) ~ R + [ o r q ( t ; Z o ) ~ R ], t ~ [ 0 , t'). •
3. Dynamic behaviour of the feedback system
Notation. (a) Any solution of the CLS (4) starting from Zo ~ R 2 will be denoted by q(t;Zo) = (x(t, Zo),y(t, Yo)). (b)
Basic features of the dynamics of CLS operating over (4) have been considered in the paper by Hejmo (1987). Here, attention will be focused on some singular phenomena that can appear in the solutions of (4). Results of theoretical analysis interpreted from a practical point of view lead to the formulation of some suggestions for technical applications.
Solutions of the systems (7) and (8) will be denoted by q+(t; z0) = (x+(t, Zo), y÷(t, Xo)) and q (t; Zo) = (x_(t, Zo), y-(t, yo)), respectively. (c) If Yl > 0 then we define T +, by T+= T ~ U T ~ where T~-={(x,y):x=h(y),y<-O}, T~= { ( x , y ) : x = h ( y ) , y ~ ( O , yl)}. But, if y , < 0 then we define T - , by T = T { U T ~ where T{={(x,y):x=h(y),y->O}, T ~ = { ( x , y ) : x = h ( y ) , y E ( y , , O ) } (Fig. 1). (d) A point intersection of the trajectory T + or T - with the x-axis will be denoted by z' = (x', 0) = (h(0), 0). •
Theorem 3.1. For a given time-optimal CLS (4) and (5), let Yl > 0 [or <0]. Then we have: (i) There exists a finite time 0 < 4 such that from each Zo E T2~ [or ET£] there starts a time-optimal, but non-unique R-solution q+(t; Zo) e T~ [or q (t: z(~) E T 2 ]. t E [ 0 , q ) , q + ( t t ; Z o ) = Z l [ o r q (tt;z~)=zl]. (ii) There exists a finite time 0 < t 2 such that from each zoe T~- [or E T2] there start non-unique R-solutions
k=y,
x(0)=xo;
f=f(y)+K,
y(0)=yo,
(7)
~=y,
x(0)=Xo;
~,=f(y)-K,
y(O)=yo.
(8)
1939
Brief Papers q (t;Zo)•R [or q + ( t ; Z o ) • R + ] , t • [ 0 , t2), q _ ( t a ; Z o ) • T~ [or q+(t2; Zo) • Ti-], which are not time-optimal ones.
are able to find a technical interpretation of the dynamics of the CLS (4) operating in real circumstances.
Proof. (i) The proof of this part results immediately from (4), (5a), (9a) [or (4), (5b), (9b)], Lemma 2.2 and Remark 2.3. (ii) From Lemma 2.2 and Remark 2.3, it follows that there exists a finite time t2 > 0 such that q (t; Zo) • R - [or q+(t;Zo)•R+], t • ( 0 , t2). But (4) implies too, that its q-solution q (t;Zo) • R - [or q÷(t; Zo) • R÷], t • (0, t2). Extending this solution to the boundary of a domain where it exists we get: q ( t ; Z o ) ~ R [or q + ( t ; z 0 ) • R ÷ ] , t • [ 0 , tz), q (t2;zo) • T ? [or q+(t2;Zo) • Ti-]. Optimal solution contained in (i) and non-optimal one in (ii) is obvious. •
Remark 4.1. If inaccuracy ( l l ) occurs, then from any initial state Zo E R2\zl there starts the trajectory of the unique q-solution which spirals around the target, zl and the point z' = (h(0), 0). This trajectory may tend either to a limit cycle (with z~ and z' in its interior) or may encircle the points z~ and z' indefinitely many times, while diverging. •
Practical identification of the real motion resistances function f ( y ) meets with difficulties. Consequently, the model of the object taken for CLS theoretical synthesis is an inaccurate mapping of this object. Hence, because of inexact identification of the function f(y), the switching curve generated by a real CLS (4) may become 'more' or 'less' steep than the time-optimal one. The above cases of the switching curve inaccuracy are defined as follows: (a) h ' ( y ) < F _ ( y )
a.e. on
[Yl, ~);
(b) h'(y)
a.c. on
(-:¢,Yl];
(a) h ' ( y ) > E ( y )
a.e. on
[yj, o~);
(b) h'(y)>F÷(y)
a.e. on
(-~,Yl].
(10) (11)
Theorem 3.2. Given CLS (4), let inaccuracy (11) hold and Yl > 0 [or <0]. Then we have: (i) There exists a finite time 0 < t l such that from each Zoe T~ [or • T ~ ] there starts a non-unique q-solution q+(t;Zo) • R ÷ [or q_(t;Zo) • R-], t • [0, tl), q+(tl;Zo) • T [or q_(fl; Zo) • T +] which is not a time-optimal one. (ii) There exists a finite time 0 < t2 such that from each Zo• T ] [or •T~-] there starts a non-unique %solution q (t;z0) • R - [or q+(t;Zo) • R+], t E [0, t2), q-(t2;Zo) ~ T~ [or q÷(t2; Zo) • Ti-] which is also not time-optimal. Proof. The proof of (i) results immediately from (4), ( l l b ) , (5a) [or (4), ( l l a ) , (5b)], Lemma 2.2 and Remark 2.3. (ii) can be proved in the same way as was done in the proof of T2 in Theorem 3.1. • Theorem 3.3. Given CLS (4), let inaccuracy (10) be valid and Yl > 0 [or <0]. Then we have: (i) From each zo • T [or • T +] there starts the unique ~-solution, the trajectory of which lies totally on T - [or T +] and reaches the target zl in finite non-minimal time. Moreover, none of the q-solutions start from Zo • T - [or • T ÷] for increasing time. (ii) From each Zo • T~-\z' [or •T~\z'] there starts the unique 5~-solution, the trajectory of which lies totally on T~[or T1] and reaches z' = (h(0), 0) in finite non-minimal time. Moreover, none of the %solutions start from Zo • T { \ z ' [or • T?\z'] for increasing time. (iii) From z' there starts a trajectory of the unique ~-solution q(t;zo)~z', t • [0, ~) and none of the qsolutions start from z' (stop of movement). (iv) There exists a finite non-minimal time 0 < t~ such that from each z~'~• T~ [or • T~] there starts a trajectory of the unique q-solution q (t;74)) • R [or q+(t;Zo) • R+], t • (0, tl) and q-(h;Zo) • T~ [or q+(tl;zo) • Ti-]. Proof. The proofs of (i), (ii) and (iii) follow the same logical pattern as that used in the proof of Theorem 3.5 in the paper by Hejmo (1987). The proof of (iv) parallels that of (ii) in Theorem 3.1. • 4. Mismatch problems The real time-optimal controller forms its own switching curve which may differ considerably from that required, e.g. due to computational errors and unknown variations in motion resistances. This switching curve inaccuracy together with errors in state identification will be called 'mismatch problems'. These may cause intolerable behaviour for the CLS (4). Referring some results of the paper by Hejmo (1987) to phenomena shown in Theoems 3.1, 3.2 and 3.3, we
Remark 4.2. Technical interpretation of the ~-solution will be done for Yl > 0 only, the case Yl < 0 being analogous. If the CLS (4) operates with inaccuracy (10) then from each z6 • T - there starts a trajectory of the unique ~-solution. On the other hand, (4) implies that from each z6 • T - there starts the q-solution q_ the trajectory of which should penetrate into R + and implies also that none of q exists in R +. Practically, the trajectory of this solution on leaving z6 • T penetrates into R* where it is immediately forced to repenetrate T (Fig. 2(A)). The real CLS (4) and (10) generates a trajectory that starts to oscillate around T - with a certain frequency and amplitude depending on the delay-time inherent in the switching operation, which evidently exists in every real structure. This trajectory of ~--solution is therefore a limit of the real oscillatory process (sliding, chattering) when the delay time tends to zero, i.e. when the frequency tends to infinity. The same interpretation may be offered for ~-solutions starting both from z6 • T~Xz' (Fig. 2(B)) and z'. The ~-solution therefore becomes a generalized description of the sliding process which has an essential practical meaning. The ~-solution is independent of whatever variation of F_(y) and F+(y) if only (10) is fulfilled. This evident fact will be used in what follows for practical switching curve formation. • Analysis similar to that done in the paper by Hejmo (1987) leads to"
Remark 4.3. Given CLS (4), (10) by y l > 0 , let us define subsidiary sets: T , = {(x, y) :x = g_(y), y >-Yl}; M - = {(x,y):h(y)
,.._A ~./"
~y /
"-L~'-.~
q+
÷
z1
© X
©
B
FIG. 2. Directions of emission of solutions.
1940
Brief Papers Trn
x
~.
~Y
©
t C
x T+, "~.
ZI3
M÷
FIG. 3. Sliding switching curve.
x" < x' = h(0)] in finite time (Fig. 3). The control process for the case y~ < 0 is analogous. • Remark 4.4. Mathematical phenomenon for the occurrence of non-unique %solutions will be interpreted for y~ > 0 only, the case y~ < 0 being analogous. In the real CLS, because of technical reasons (Hejmo, 1987) the states which belong exactly to the switching curve T~ may be identified by the control system as belonging to the region R - . So, the CLS (4) attributes the control function u = - 1 to these states, that generates the ~-solution q the trajectory of which penetrates into R - and afterwards intersects the switching curve Ti ~ in the point ~ • T~\z'. If the switching curve satisfies (5) [or (10)] then from z7 there starts the %solution [or ~-solution] which transfers the state in finite time along T~ to z'. (a) If (5) occurs, then from z' there starts the ~-solution, the trajectory of which transfers the state along T~ and may bring it in finite time to the same point Zo • T2~ in which the movement of the CLS began. If Zo • T2~ is identified, with even the least systematical error, as belonging to the set R , then the CLS (4) generates the limit-cycle a limit-curve of which contains z' in its interior and does not contain z~. Thus, this CLS becomes unable to reach the target z~ (Fig. 4). (b) If inaccuracy (10) occurs then z' becomes an 'end point' of the movement. The CLS (4) is stopped there, because the unique if-solution remains in z' in an infinite time interval. •
Ti FIG. 5. Errors in state measurement.
may recognize the state z~ = (x~., y~) ~ T as belonging to T. Thus, in zE the switching operation is executed. Let the CLS (4) identify the co-ordinates of the state with an error p > 0, to > 0 such that (x~ -4-p, y~ ± to) • T. We will consider a case y ~ > 0 only. Let T and T~ satisfy (10) but TJ do (11). (a) From each z,. such that ( x ~ . - p , y , - t o ) • T and ( x , + 0, Y~ + to) • T - there start, respectively, the trajectories Tm and Tp of the unique ~-solutions which are not able to reach z~. These trajectories can reach only the states forming a certain neighbourhood of zt. The smallest one is a ball B(zl, e) where radius e = (p2 + to2)u~ Trajectories T,, and Tp reach in finite time the boundary B of the B(z~, e) in the points z ~ = (xl + p , yl + to) • B and Zp = (xl - p , yl - to) • B, respectively (Fig. 5). However, from each z, such that (x~ - p, yE - to) • T/- and (x~ + p, y~. + to) • T~" there start, respectively, the trajectories Ti~,,, and T{t, of the unique .~-solutions that reach in finite time points z., = (x' - p, - t o ) and zt~ = (x' + p, to) identified by the CLS as z' belonging to the x-axis (Fig. 5). (b) From each z,- such that (x~-p,y~-to)•T~ and ( x ~ + p , y ~ + t o ) • T ~ there start, respectively, the trajectories T~m and T~p of the unique qg-solutions q+ which start from z " and Zp and reach in finite time z + = ( x l + p , y l - t o ) • / ~ and z ~ = ( x l - p , y ~ + t o ) • / ~
Remark 4.5. The CLS (4) operating in real circumstances
,y - . . Tm
® T2p/1¢//' # ¢#'¢"
T2 l I
/
+ T2m
+
\ FIG. 4. Non-unique solutions.
FIG. 6. Target set and the state trajectories.
Bdef Papers respectively. It is obvious that from each z~ ¢ T ~ there starts the trajectory of the unique C~-solution q+, which runs across R + and intersects T - in a finite time. From z, e T~o there starts the trajectory of the unique qg-solution q_+ reaching in a finite time either T - or B. Which set ( T - or B) is reached by the trajectory of that Cg-solution q+ depends on the values of the state measurement errors p and to (Fig. 6). •
5. Concluding remarks and suggestions for applications In practice, the errors p, to depend on the technical means applied in control structure. Remarks 4.1-4.5 imply the following principles of CLS synthesis. (1) We replace zl by the target set B after estimating maximal values of p, oJ. (2) The shape of T - [or T +] and T~- [or Ti-] must satisfy (10) but T~ [or T~] must satisfy (11) for all possible variations in resistance function
f(Y). For each Zo e R2\B the CLS synthesized in accordance with the above principles ensure the target set reaching along the unique trajectory in finite time independent of variations in motion resistances. The time it takes to reach the target differs from the minimal one, and the difference depends on how much the real switching curve differs from the
1941
time-optimal one. Besides, the control process is as close as possible to the optimal one. Examples in the work by Hejmo and Kloch (1989) show a comparison of the time-optimal process with the suboptimal one generated by CLS operating with an inaccurate switching curve.
References Brunovsky, P. (1974). The closed-loop time-optimal control. SIAM J. Control, 12, 624-634. Filippov, A. F. (1964). Differential equations with discontinuous right-hand side. Trans. Am. Math. Soc. 42, 199-231. Hajek, O. (1979). Discontinuous differential equations. J. Differ. Eqns, 31, 149-185. Hejmo, W. (1987). Stability of a time-optimal closed-loop system with parameter changes. Int. J. Control, 45, 1161-1178. Hejmo, W. and J. Kioch (1984). On the time-optimal problem of positional control with discontinuous resistances of motion. RAlRO-System Analysis Control, 18, 329-341. Hejmo, W. and J. Kloch (1989). On the perturbations in a time-optimal closed-loop system. Ann. Polonici Mathematici. 50, 37-52.