- Email: [email protected]
Constrained stabilization of continuous-time linear systems
$)rST|,qltS f. CONlltOt. UTNItS ELSEVIER
Systems & Control Letters 28 (1996) 95-102
Constrained stabilization of continuous-time linear systems Franco Blanchini a,,, Stefano Mianib aDipartirnento di Matematica ed Inforrnatica, Universit3 degli Studi di Udine, Via delle Scienze 208, 33100 Udine, Italy bDipartimento di Elettronica e Informatica, Universit3 degli Studi di Padova, via Gradenigo 61a, 35131 Padova, Italy
Received 15 January 1995; revised 23 May 1995
Abstract
In this paper we consider the stabilization problem for linear continuous-time systems, under state and control constraints. We show that the largest domain of attraction to the origin can be arbitrarily closely approximated by a polyhedral domain of attraction associated to a certain (continuous) feedback stabilizing control and we show how to use existing numeric procedures for discrete-time systems to solve the continuous-time problem. We propose a new discontinuous stabilizing control law for scalar-input systems which has the advantage of being successfully applicable to systems with quantized control. Keywords." Constrained control; Linear systems; Positively invariant sets; Non-linear control
1. Introduction
It is well known that in practical problems the presence o f constraints on the state and control variables introduces severe restrictions on the choice o f a feedback control system. Exceeding the prescribed bounds may be either dangerous or undesirable or even impossible. Several authors devoted their attention to the development o f synthesis methods which enable to take into account time-domain constraints during the control design. Particularly, efficient methods are those based on the construction of invariant sets [1, 5-11, 14, 16]. Although the mentioned references consider different particular problems, they have as a common denominator the following fundamental idea: to assure that no constraints violation occurs they seek for a control which makes positively invariant a certain subset o f the admissible state set and assumes admissible input values in such a set.
* Corresponding author. Fax: 39 432 558499; e-mail: BLANCHINI@ UNIUD.IT.
It is known that if the constraint set contains the origin in its interior and a stabilizing control exists, there also exists a domain o f attraction to the origin where no constraint violations occur. However, this set must be sufficiently large to be satisfactory and one o f the criteria which can be considered for control design is that of "maximizing" its size. In this spirit, [8, 10] provided computational methods to approximate the "largest" domain o f attraction included in an assigned compact state constraint set for a discrete-time constrained system and derived feedback stabilizing controls. However, these methods have no continuoustime counterparts. The contribution o f this paper is twofold. In Section 2 we show how to approximate with arbitrary precision the largest domain o f attraction of a continuoustime system with a polyhedral domain of attraction. In Section 3 we show that the provided domain o f attraction can be associated to a variable structure (Lipschitz) controller already presented in literature. Then we provide a new variable-structure f e e d b a c k control law for scalar input systems which is no longer continuous but has the advantage of being much
0167-6911/96/$12.00 Copyright (~ 1996 Elsevier Science B.V. All rights reserved PH S01 67-691 1(96)000 13-8
96
F. Blanchini, S. Miani/ Svstems & Control Letters 28 (1996) 95.-102
less complex for on-line implementation. This control is of particular significance to solve problems in which the control is quantized. We briefly discuss the robustness of the compensator in Section 3.1. In Section 4 we show an example of application of the new control to a DC-electric motor. In Section 5 we give some concluding remarks.
2. Approximation of the largest domain of attraction With the term C-set we denote a convex and compact set containing the origin as an interior point. It is known that any C-set P induces a positively homogeneous function which is known as Minkowski functional
Tp(X)
=
inf{). > O: xE)~P}.
If P is O-symmetric, (i.e. x E P implies - x E P ) then Up is a norm. Consider the following linear system: .f(t) = Ax(t) + Bu(t),
( 1)
where the system state x ( t ) ~ ~ and control input u(t) E ~q are subject to the constraints x(t) E X,
(2)
u(t) E U,
(3)
where X C ~ and U C Eq are assigned polyhedral Cset. Due to the presence of the constraints, the solution of the stabilization problem implies restrictions on the admissible initial conditions. Definition 2.1. The C-set S C X is a domain of attraction (with speed of convergence t ) for system (1) if there exists fl > 0 such that for all x0 E S there exists a piecewise continuous control function u(.) : R -+ U such that the trajectory x(t) with initial condition x(0) = xo corresponding to u(t) is such that
partial ordering since if $1 C X and $2 c X are two domains of attraction for some t , then their convex hull is also a domain of attraction. This property implies the existence of the "largest" domain of attraction S& Ideally, given fl as performance specification, we would like to find a feedback control assuring such a speed of convergence for all the initial states in S[~. In fact, we will solve the suboptimal problem of assuring a speed of convergence [3 arbitrarily close to/3 for all the initial conditions belonging to a polyhedral set P which fits with arbitrary precision S~. To this aim, we consider the Euler approximating system (EAS) of (1) which is the following discretetime system: x(t+l)=[l+zA]x(t)+rBu(t),
r>0.
(5)
An obvious extension of the Definition 2.1 holds for the discrete-time system (5). Definition 2.2. We say that the set S is a domain of attraction (or a )L-contractive set) for (5) if there exists ;o < l such that for all x0 E S there exists a sequence u( k ) E U such that Us (x(t)) ~ Us(x(O));/. We show that the maximal domain of t-attraction for (1) can be arbitrarily closely approximated by a polyhedral domain of attraction for (5) which is still a domain of attraction for the continuous-time system with a speed of convergence arbitrarily close to ft.
Lemma 2.1. Assume that there exists a C-set S which is a domain o f attraction f o r (1) with a speed o f convergence fl > O. Then f o r all fl' < fl there exists ~ > 0 such that the set S is contractive f o r the EA S with 2 ~ =- 1 - zfl t. Proof. Condition (4) implies that for everyx E S there exists u such that D + ( x , u) --
lim sup
T s (x + z(Ax + Bu)) - T s (x)
~'~0
Ts (x(t)) <<.e ~tUs (x(0)).
(4)
If we take fl = 0, the set S is simply said to be Uinvariant [8]. We say that S/~ C X is the largest domain of attraction if for any domain of attraction S in X with speed of convergence fl we have S C S/~. It is immediate to see that the domains of attraction in X are subject to a
~< - f l U s (x).
(6)
Then for each ff
Us(X)
<~ - ~ ' U s ( x ) (7)
F. Blanchini, S. MianilSystems & Control Letters 28 (1996) 95-102
We prove now that this property is uniform in x: for each/3' < fl there exists ~ such that for each x ~ S there exists u such that (7) holds. By contradiction assume that there exist the sequences {xk ~ S} and rk ~ 0 such that
97
Our goal is to find a domain o f attraction i n X which is as close as possible to the largest one. We say that the C-set P internally approximates in X the C-set S with precision e > 0 if (1 - e ) S C P C X .
tPs (xk 4, zk(Axk + Bu)) - q~s (x~)
> -/~' q's (xe)
-rk
for all {u ~ U}. Since S is a compact set, the sequence {xk} admits a converging subsequence. Without restriction we assume that xk + 5. Let/~" such that -/3 < - fl" < -/~'. There exists "~and fi such that
qJs (54- £(A£ 4, Bfi)) - 7Js (5)
(9)
The next theorem, which is the main result of this section, states that we can internally approximate with arbitrary precision the largest domain o f attraction SI~ with a polyhedral domain of attraction P loosing an arbitrarily small amount in terms of speed of convergence. Theorem 2.1. Let S~ the largest domain of attrac-
~< -/3" q's(,~).
Now the difference quotient is a non-decreasing function of z if q~s is convex [13] then for zk ~<
~Ps (xk 4. zk(Axk 4. Bgt)) - 7*s (xk) ~k
tion in X for (1) with speed of convergence fl > O. Then for all 0 0 there exists a polyhedral C-set P such that (1 - e ) S ~ C P C X and such that P is a domain of attraction for (1) with speed of convergence /3. Moreover, the control can be expressed in a feedback form u = Cb(x), where 4~ is a Lipschitz.function on P.
<~ qJs (xk + i(Axk + Bfi)) - 7~s (xk) = ~s(Y:4,{(A54.Bfi))-{
7*s(~) 4.^l(5'(xk), "C
where in view of the continuity of ~s, (5'(xk) --+ 0 as xk --+ 2. So for sufficiently large k
qJs(xk 4, zk(Axk 4. B~)) - ~s(xk) "r,k
<. -/3'~s(;),
a contradiction. Therefore there exists -g such that for all x there exists u assuring (7). Note that it is not restrictive to assume that f is such that 1 - -gfl' = 2' < 1. Then
~Ps ([I 4- fA]x 4. ~Bu) <~ [1 - f/3']~s (x) = 2'Ws (x).
(8) This immediately implies that S is 2'-contractive for the EAS. Lemma 2.2. Assume that the C-set P is ),'-contrac-
tive for the E A S for some 2' such that 0 <~;t' < 1. Then P is a domain of attraction for (1) with /3' = (1 - )o')/z. Moreover, there exists a Lipschitz feedback control function q): P ~ U assuring condition (4). Proof. See [2] or [4].
[]
Proof. Take/~ < / 3 ' < fi and fix e. > 0. From Lemma 2.1 we can find z such that S/~ is contractive for the EAS with )~' = 1 - ~/3' < 1. Consider the largest )~'-contractive set R;, included in X, then S/~ C R;,. Let 2 = 1 - z f l > 2 ' , from [3], Theorem 3.2, R~, can be arbitrarily closely approximated by a polyhedral )w-contractive set P C X in the sense that (1 - e ) R ; , C P C R ; , . Now S~CR~, implies (1 - e)S/~C(1 - e.)R;, C P C X . From Lemma 2.2, the polyhedron P is a domain of attraction with speed/3 for the continuous system. The existence of a (Lipschitz) feedback control follows from [2, 4]. [] From the theorem above we have that to approximate the maximal domain of attraction with a polyhedral domain o f attraction, we have just to use the EAS with a sufficiently small ~ and to apply the method proposed in [3] to approximate the largest 2 contractive set for the EAS. The derived polyhedron is a domain of attraction for ( 1 ). Moreover, note that a set is 2-contractive for the EAS if and only if it is invariant for the following modified Euler approximating systems
x(t + 1) = [I + ZA]x(t ) Jr- =-u(t). zB A
(10)
A
This means that we can also apply the procedures proposed in [8, 10] for discrete-time system.
98
[( Blanchini, S. Miani/Systems & Control Letters 28 (1996) 95 102 1
0.8 0.6 0.4 0.2
o -0.2 -0.4 -0.6
-0.8
"'-1
-0.8
-0.6
-01,4
-0'.2
¢2
0'4
0~.6
0.8
Fig. 1. EAS (dotted) and exponential (dashed) approximation of PcoN.
Remark2.1. An important question is whether we can determine an approximation of the largest domain of attraction via exponential approximation. The answer is affirmative with a fundamental difference: the derived approximating set is not in general a
the continuous time system but it is invariant as well.
3. Feedback control
domain o f attraction f o r the continuous-time system.
In Fig. 1, we show the largest invariant set PCON C X, x -- {x: [x,I ~< 1}, u -- {u: lul ~< 1} for a double integrator system with 1
(in this simple case PCON is very easily derived). In Fig. 1 we show PEAS C X , the (internal) approximation obtained through the EAS and the polyhedral approximation obtained with the exponential approximation PEXP C X using z = T = 0.5. Since PEXP ~ PcoN, PEXP is not invariant (this is confirmed by checking vertex conditions). This limit of the exponential approximation was also mentioned in [15] where the problem of the construction of the largest invariant set contained in X for a stable linear system was considered. As a particular case with B = 0 and A stable, we solve that problem by providing a set which not only converges to the largest domain of attraction for
We assume now that a polyhedral domain of attraction P is known and we consider the problem of associating to P a feedback control. A suitable stabilizing control is the linear variable-structure one proposed in [3, 8] as follows. We associate to each vertex x of P, a control u E U such that (6) holds. Each of these control vectors "pushes" the vertex state inside the polytope P. Then we divide the polytope P into simplicial sectors Pi, each of them formed by the origin and n distinct vertices of P. According to [3, 8], the partition in sectors must be such that/oh has a nonempty interior, Ph n Pk has empty interior if k ¢: h, and [,-JkPk = P. Techniques to obtain this partition are discussed in [12]. In the sector h the linear gain u = Kh x, with Kh = Uh X~- i
has to be applied, where Xh is the matrix formed by the n non-zero vertices of Ph and Uh is the matrix of the corresponding control vectors. In [4] it is shown that this control is Lipschitz and assures (6) on P.
F, Blanch&i, X Miani/Systems & Control Letters 28 (1996) 95-102
The considered linear variable-structure control is easy to apply as long as the system dimension is small. If this is not the case, computational problems may arise due to the number o f simplicial sectors Ph which can be very high. We propose, for single input systems, a new compensator which is discontinuous. The advantage of this compensator is twofold: first it is less complex than the previous, second it is amenable for systems with quantized control since it takes values on the boundary of U. Assume that the C-set P has the following plane representation:
P = {xE~n: Fix <~ 1, i = 1..... an} = {xE ~n: Fx<<.T}.
( l 2)
Define now I(x) as the set of all indices where the maximum in (12) is reached (clearly this set is a singleton i f x is in the interior of a sector)
l(x) = {i: i = argmaxj Fix}.
The Lyapunov derivative for this function is given by
D+(x, u) = lim sup
qJp (x + ~(Ax + Bu)) - qJp (x)
r--+0
= max Fk(Ax +Bu). kEl(x)
(15)
By assumption for each x E P there exists u ~ U such that D+ (x, u) <~ - fl~p (x ). Now, to enforce maximal contractivity we can try to find a controller u = q~(x) by solving the optimization problem of minimizing D+(x,u) with respect to u. For all x in the interior int{Sk} of a sector Sk we have I(x) = {k}, then the solution of the minimization problem inf Fk(Ax + Bu) = FkAx ÷ inf (FkBu)
(11)
The polytope P can be divided in "natural" sectors Si which are the pyramids each o f them obtained as the convex hull of the origin and one facet (i.e. an (n - 1 )dimensional face) of the polytope. Given x E P, the sector including x is that associated to the ith facet, where i = argmaxj F i x.
99
(13)
Note that for a general polyhedral C-set P, the natural sectors are different from the simplicial sectors. For instance in a three-dimensional cube each square facet generates one natural sector and two simplicial sectors (which are tetrahedra). The number as of simplicial sectors is clearly greater or equal to the number an of natural sectors (the polytopes having the property an = as are called simplicial) and the ratio between the two numbers can grow exponentially with the space dimension. The main idea of the new compensator is to try to maximize the speed of convergence to the origin measured with respect to the norm induced by P. As a result, we will derive a compensator which is constant in each natural sector. Henceforth, we assume that q = 1 and that the control is constrained as u c U = [u-, u+].
uEU
uCU
is reached on the same extremum of U. Denote by u k E { u - , u +} such a solution, then the following variable-structure (bang-bang) control can be defined u = ~ B ( x ) = Uk for x r S k ,
(16)
which is constant in the interior of each sector. Since the definition of the control ~b~ is ambiguous in the intersection of several sectors associated to different controls, ambiguity is removed by arbitrarily assigning to any x which belongs to such an intersection one of the extreme values of U. It is obvious that as long as the trajectory of the system with this control evolves in the interior of the sector the condition (4) is assured. However this control may not be continuous on the inter-sector boundary and it may introduce sliding trajectories. Nevertheless, this controller assures global convergence according to the following theorem.
Theorem 3.1. Assume that the polyhedral C-set P & a domain of attraction for (1) with speed of convergence ft. Then the control q~BBdefined as (16) assures convergence with a speed fl Jor all initial conditions inP. Proof. To prove the theorem it is sufficient to prove that u = 4~BB(X) assures (6) for all x in P. The condition (6) is obviously assured i f x is in the interior of a sector. Thus, we assume that x is on an inter-sector boundary ~ individuated by Fkx = Up(x), k E I~. Define as Z the smallest subspace including ~ . Z is defined by the equalities
If P is represented as in (11 ), then the induced norm is (see for instance [16])
Fix = Fjx,
q~e(X) = max Fix.
Without restriction assume that the control is not continuous on .~ (say it takes different values on the
i--1,...,an
(14)
i,j E I~.
F Blanchini, S. Miani/Svstems & Control Letters" 28 (1996) 95 102
100
sectors incurring in ~ ) . If the state leaves .~ then the decreasing condition (6) is fulfilled and thus let us consider the case in which the motion lies on ~ for t in some nonzero interval [t0, tl). Invoking the equivalent control method [17], there must exist U~q such that u - ~< ~/eq ~ u+ and that Ax + Bueq is tangent to 27. In our case, being 2; a linear manifold, this is equivalent to Ax 4- Bueq C ~ which implies that F j ( A x 4- Bueq ) =
for some positive ft. In this case, as the state approaches the origin, we can reduce the control effort by jumping between intermediate points, say uE {-(k/r)a,(k/r)g},
k E {0, 1. . . . . r}
instead of between -~7 and ~7. The value of k can be chosen in such a way the condition D+(x,u)<~ - fTJ(x) is assured for [u[ ~< (k/r)~. By linearity it is immediate to show that, denoting by Ikt] the smallest integer greater or equal to It E R, k can be chosen as
Fi(Ax + BUeq ) f o r i,j E 1~. k(x) = FrqJ(x)].
To prove the theorem, we need to show that the equivalent control assures (6). By contradiction assume that this is not the case, then for t E [to, tl ), D*(x, Ueq ) • -- [~tlJp(X). From (15) D + ( x , Ueq) = F j ( A x 4 - B u e q ) > - [;]Up(x)
for j E / e .
(17)
By construction of P for every x there exists u* such that D+(x,u *) = max F/(Ax + Bu*) <~ - fillip(X). /
(18)
Now one of the two inequalities u*~< Ueq ~< u +, /2-- ~ Ueq ~ b/* holds. Assume u* ~< Ueq ~ U+ (the other case is obviously equivalent). For every j E f~, by the linearity of Fj(Ax +Bu), from (1 7) and ( 18) we get Fj(Ax + Bu +) > - f ~ p ( X ) for all j E/~e. Hence for all concurring sectors Sj, j E I~, the minimizing control 4~nB(X) must be equal to u - in contradiction with the assumption that the control is discontinuous on/~.
[]
As a final comment, we remark that in practical applications, the control passes through a sampling device. So, imposing to the control to take values on the extrema may cause considerable inter-sampling ripples which may be unacceptable when the state approaches the origin. This problem can be overcome by reducing the sampling time. Sampling-time reduction can be avoided if the quantized actuator can take values not only at the extrema but in a finite number of points. Assume for simplicity that U is no more an interval but a set of the form
R e m a r k 3.1. To eliminate ripples we can proceed in a different way by "switching" to a linear compensator u - K x when the state reaches a sufficiently small neighbourhood V of the origin (clearly this method does not work with quantized control). Our techniques, applied as particular case can be used to approximate the largest invariant set in V for the closed-loop system 2 = (A + BK)x by an invariant polyhedral set V*. Thus "switching" can be safely done as soon as the condition x(t) E V* is verified. Note that in this way we can improve the asymptotic behavior of the system by properly assigning the eigenvalues o f A + BK. R e m a r k 3.2. The definition of the control q~B8 can be extended to the case of multi-input systems. In this case the control assumes values on a vertex of U in each sector, but it can be easily shown that nonconvergent sliding trajectories may arise in the intersector boundary. R e m a r k 3.3. The sampling time T used to practically implement the control, has not to be equal to the parameter r we use to compute the set. In practice we should have T << r. 3.1. Robustness o f the compensator This paper together with the previous references [2-4] furnishes all the material to provide a design method which not only takes into account state and control constraints, but derives a control which is robust against structured parametric perturbations. Assume that A and B are uncertain matrices of the form r
A(w) = ~ A i w i , i--1 '
U = {-ti,
(r-1)£,r
! ~ 7 , 0 , ! f f . . . . . ~7}
B(w) = E B i w i , i 1
wi • i=1
1,
w i • O.
(19)
F. Blanchini, S. Miani/ Systems & Control Letters 28 (1996) 95-102
101
Tablel
~]
~2
1 2
1.0000e + 0 4.3805e -
0.0000e + 0
0.0000e + 0
12
1.0562e + 0
1.1072e + 1
1.6853e -- 0
3 4 5
6.9169e 6.9806e 1.2750e Y1260e 7.4470e 5.0536e 6.3672e
2.4989e 2.3446e 1.9799e 2.1248e 2.6163e 2.5717e 2.8315e 2.6869e 2.3492e
13 14 15 16 17 18 19 20 21
2.1837e 6.8219e 4.1632e 1.0444e 6.0438e 6.5936e 5.7590e 1.0204e 8.5464e
6.0240e 1.2939e 9.6618e 1.0573e 8.7796e 1.2310e 1.2281e 1.0063e 1.0610e
2.4156e -- 0 1.6666e + 0
6 7 8 9
7.3046e 1.3776e 1.3515e 4.2319e 7.7821e 9.891 le 1.1183e 1.3210e 1.1186e
k
10 11
-
9.4652e 1.0359e + 0
~3
+ + + ÷ + + + + +
0 1 1 0 0 0 I 1 1
1.1398e + 1
k
+ + -+ + + ÷ ÷ +
0 0 0 0 0 0 0 0 0
~l
2.0107e + 0
~2
--+ + -
I 1 1 0 1 1 1 0 1
--
~3
+ + ÷ + + + + + +
0 1 0 1 0 1 1 1 1
2.7826e ÷ 0 1.4167e + 0 2.5864e + 0 1.4145e + 0 2.7646e + 0 1.2055e + 0 2.4594e + 0
--
0.4-, 0.3~ 0.2~ 0.1
i :
i .
-::.5.
i "
"
"-. " . .........
.....
i ..
i... ....
-
0
.....
.
0
......
4
". "
...............
•
......
: i
..
..
.......
~
o.2
-0.2
-0.06 -0.8 Fig. 2. The simulated trajectory with sampling time T = 10 - 3 .
As shown in [3] we can form a polyhedral 2contractive polyhedral set for the EAS which is a domain of attraction for the continuous-time system [4]. As long as the control is scalar, the proposed bang-bang compensator can easily be extended by taking in the sector Sk the solution ~BB(x) -- uk of the following problem: min
u- <~u.<.u+
m a x F k ( A i x + Biu).
Proving the state convergence for the closed-loop system is a simple extension of Theorem 3.1, and we skip the proof for brevity.
4. Example Consider the following continuous-time 3-dimensional model of an armature-controlled DC electric motor. We assume that Xl is the armature current, x2 is the angular speed and x3 is the angle. The control u is the armature voltage. This is a typical example of a system in which the control quantization is an important aspect. Indeed voltage suppliers are very often able to provide values only on a finite set. So assume that the control can take values in the finite normalized set { - 1, -0.5, 0, 0.5, 1}. The state values are normalized with respect to the maximal allowable
102
F Blanchini, S. Miani/ Svstems & Control Letters 28 (1996) 95 -102
values so that X = {x E N~: A and B are
A =
-0.0700 0.0600
-0.8600 0.0085
o.oooo
Ixil ~ 1 }.
The matrices
0.0000] / 0.0000[
.oooo o.ooooj
F l.OOOOl B = /°°°0°/
LO.OOOO_l A p p l y i n g the procedure in [3] with r : 1 and ,:. = 0.8 we derived the polyhedral set P = {x" - 1 ~< F i x ~< 1, i = 1,2 . . . . . 21} w h i c h is 0 - s y m m e t r i c since X and U are both symmetric. The rows o f the matrix F are reported in Table 1. A c c o r d i n g l y to L e m m a 2.2 we have that c o n v e r g e n c e is assured with [3 = 0.2. Since the set S is symmetric each sector is associated to its opposite one. Using the linear variable structure controller we have to consider 124 simplicial sectors in each o f w h o m we apply a linear control. Using the control tbBB we have just to consider 42 natural sectors. W e denote as positive sector the one associated to the facet F k x 1 and as negative, the one associated to the facet F k x = - 1 . Since Fk can be exchanged with - F k , in Table 1 we report Fk in such a w a y that the corresponding positive sector is associated to u = - 1 ( o f course the negative sector is associated to u = +1 ). Finally, in Fig. 2 we report the 3-D simulated state trajectory from the initial point x ( 0 ) = [0 0 0.3532] r, which has been obtained by using the quantized b a n g bang control with a sampling time T = 10 -3 for an interval o f 24 s (the angle 0.3532 corresponds to the state with m a x i m u m angle and null initial speed and current belonging to P).
5. Conclusions In this paper it was shown h o w to approximate the m a x i m a l d o m a i n o f attraction for a linear continuoustime system with state and control constraints via Euler A p p r o x i m a t i n g System. W e p r o v e d that the approximating set is a domain o f attraction for the continuous-time system while this property is not assured by domains determined through the classical exponential approximation. This domain o f attraction can be associated to a continuous feedback control. For the single-input case we p r o v i d e d a variable structure controller which is no m o r e continuous, but
has the advantage o f requiring less computational effort than the previous one and being suitable for applications to quantized input systems.
References [1] A. Benzaouia and C. Burgat, The regulator problem for a class of linear systems with constrained control, Systems Control Lett. 10 (1988) 357-363. [2] F. Blanchini, Ultimate boundedness control for discrete-time uncertain system via set-induced Lyapunov functions, 30 ° IEEE Conj'. on Decision and Control, Brighton, United Kingdom (December t l 13, 1991) 1755-1760. [3] F. Blanchini, Ultimate boundedness control for discrete-time uncertain system via set-induced Lyapunov functions, IEEE Trans. Autom. Control AC-39 (1994) 428-433. [4] F. Blanchini, Non-quadratic Lyapunov function for robust control, A utomatica 31 ( 1995 ) 451-461. [5] F. Blanchini, F. Mesquine and S. Miani, Constrained stabilization with assigned initial condition set, Int. J. Control 62 (1995) 601-61% [6] C. Burgat and S. Tarbouriech, Positively invariant sets for constrained continuous-time systems with cone properties, IEEE Trans. Autom. Control 39 (1994). [7] E.B. Castelan and J.C. Hennet, Eigenstructure assignment for state constrained linear continuous time systems, A utomatica 28 (1992) 605-611. [8] P.O. Gutman and M. Cwikel, Admissible sets and feedback control for discrete-time linear systems with bounded control and states, IEEE Trans. Autom. Control AC-31 (1986) 373 376. [9] P.O. Gutman and P. Hagander, A new design of constrained controllers for linear systems, IEEE Trans. Autom. Control AC-30 (1985) 22-33. [10] S.S. Keerthi and E.G Gilbert, Computation of minimum-time feedback control laws for discrete-time systems with state and control constraints, IEEE Trans. Autom. Control AC-32 (1987)432 435. [11] J.B. Lasserre, Reachable, controllable sets and stabilizing control of constrained systems, Automatica 29 (1993) 531 -536. [12] S. Miani, Tecniche di sintesi del controllo per sistemi con vincoli su statie ingressi, Thesis, Universit~idi Padova, 1993. [13] R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1972). [14] M. Sznaier and M.J. Damborg, Heuristically enhanced feedback control of constrained discrete-time systems, Automatica 26 (1990) 521-532. [15] K. Tan and E. Gilbert, Linear systems with state and control constraints: the theory and the applications of the maximal output admissible sets, IEEE Trans. Autom. Control 36 ( 1991 ) 1008-1020. [16] M. Vassilaki and G. Bitsoris, Constrained regulation of linear continuous-time dynamical systems, Systems Control Lett. 13 (1989) 247--252. [17] V.I. Utkin, Slidiny Modes and Their Application in Variablestructure Systems (M.I.R., Moskow, 1978).
Systems & Control Letters 28 (1996) 95-102
Constrained stabilization of continuous-time linear systems Franco Blanchini a,,, Stefano Mianib aDipartirnento di Matematica ed Inforrnatica, Universit3 degli Studi di Udine, Via delle Scienze 208, 33100 Udine, Italy bDipartimento di Elettronica e Informatica, Universit3 degli Studi di Padova, via Gradenigo 61a, 35131 Padova, Italy
Received 15 January 1995; revised 23 May 1995
Abstract
In this paper we consider the stabilization problem for linear continuous-time systems, under state and control constraints. We show that the largest domain of attraction to the origin can be arbitrarily closely approximated by a polyhedral domain of attraction associated to a certain (continuous) feedback stabilizing control and we show how to use existing numeric procedures for discrete-time systems to solve the continuous-time problem. We propose a new discontinuous stabilizing control law for scalar-input systems which has the advantage of being successfully applicable to systems with quantized control. Keywords." Constrained control; Linear systems; Positively invariant sets; Non-linear control
1. Introduction
It is well known that in practical problems the presence o f constraints on the state and control variables introduces severe restrictions on the choice o f a feedback control system. Exceeding the prescribed bounds may be either dangerous or undesirable or even impossible. Several authors devoted their attention to the development o f synthesis methods which enable to take into account time-domain constraints during the control design. Particularly, efficient methods are those based on the construction of invariant sets [1, 5-11, 14, 16]. Although the mentioned references consider different particular problems, they have as a common denominator the following fundamental idea: to assure that no constraints violation occurs they seek for a control which makes positively invariant a certain subset o f the admissible state set and assumes admissible input values in such a set.
* Corresponding author. Fax: 39 432 558499; e-mail: BLANCHINI@ UNIUD.IT.
It is known that if the constraint set contains the origin in its interior and a stabilizing control exists, there also exists a domain o f attraction to the origin where no constraint violations occur. However, this set must be sufficiently large to be satisfactory and one o f the criteria which can be considered for control design is that of "maximizing" its size. In this spirit, [8, 10] provided computational methods to approximate the "largest" domain o f attraction included in an assigned compact state constraint set for a discrete-time constrained system and derived feedback stabilizing controls. However, these methods have no continuoustime counterparts. The contribution o f this paper is twofold. In Section 2 we show how to approximate with arbitrary precision the largest domain o f attraction of a continuoustime system with a polyhedral domain of attraction. In Section 3 we show that the provided domain o f attraction can be associated to a variable structure (Lipschitz) controller already presented in literature. Then we provide a new variable-structure f e e d b a c k control law for scalar input systems which is no longer continuous but has the advantage of being much
0167-6911/96/$12.00 Copyright (~ 1996 Elsevier Science B.V. All rights reserved PH S01 67-691 1(96)000 13-8
96
F. Blanchini, S. Miani/ Svstems & Control Letters 28 (1996) 95.-102
less complex for on-line implementation. This control is of particular significance to solve problems in which the control is quantized. We briefly discuss the robustness of the compensator in Section 3.1. In Section 4 we show an example of application of the new control to a DC-electric motor. In Section 5 we give some concluding remarks.
2. Approximation of the largest domain of attraction With the term C-set we denote a convex and compact set containing the origin as an interior point. It is known that any C-set P induces a positively homogeneous function which is known as Minkowski functional
Tp(X)
=
inf{). > O: xE)~P}.
If P is O-symmetric, (i.e. x E P implies - x E P ) then Up is a norm. Consider the following linear system: .f(t) = Ax(t) + Bu(t),
( 1)
where the system state x ( t ) ~ ~ and control input u(t) E ~q are subject to the constraints x(t) E X,
(2)
u(t) E U,
(3)
where X C ~ and U C Eq are assigned polyhedral Cset. Due to the presence of the constraints, the solution of the stabilization problem implies restrictions on the admissible initial conditions. Definition 2.1. The C-set S C X is a domain of attraction (with speed of convergence t ) for system (1) if there exists fl > 0 such that for all x0 E S there exists a piecewise continuous control function u(.) : R -+ U such that the trajectory x(t) with initial condition x(0) = xo corresponding to u(t) is such that
partial ordering since if $1 C X and $2 c X are two domains of attraction for some t , then their convex hull is also a domain of attraction. This property implies the existence of the "largest" domain of attraction S& Ideally, given fl as performance specification, we would like to find a feedback control assuring such a speed of convergence for all the initial states in S[~. In fact, we will solve the suboptimal problem of assuring a speed of convergence [3 arbitrarily close to/3 for all the initial conditions belonging to a polyhedral set P which fits with arbitrary precision S~. To this aim, we consider the Euler approximating system (EAS) of (1) which is the following discretetime system: x(t+l)=[l+zA]x(t)+rBu(t),
r>0.
(5)
An obvious extension of the Definition 2.1 holds for the discrete-time system (5). Definition 2.2. We say that the set S is a domain of attraction (or a )L-contractive set) for (5) if there exists ;o < l such that for all x0 E S there exists a sequence u( k ) E U such that Us (x(t)) ~ Us(x(O));/. We show that the maximal domain of t-attraction for (1) can be arbitrarily closely approximated by a polyhedral domain of attraction for (5) which is still a domain of attraction for the continuous-time system with a speed of convergence arbitrarily close to ft.
Lemma 2.1. Assume that there exists a C-set S which is a domain o f attraction f o r (1) with a speed o f convergence fl > O. Then f o r all fl' < fl there exists ~ > 0 such that the set S is contractive f o r the EA S with 2 ~ =- 1 - zfl t. Proof. Condition (4) implies that for everyx E S there exists u such that D + ( x , u) --
lim sup
T s (x + z(Ax + Bu)) - T s (x)
~'~0
Ts (x(t)) <<.e ~tUs (x(0)).
(4)
If we take fl = 0, the set S is simply said to be Uinvariant [8]. We say that S/~ C X is the largest domain of attraction if for any domain of attraction S in X with speed of convergence fl we have S C S/~. It is immediate to see that the domains of attraction in X are subject to a
~< - f l U s (x).
(6)
Then for each ff
Us(X)
<~ - ~ ' U s ( x ) (7)
F. Blanchini, S. MianilSystems & Control Letters 28 (1996) 95-102
We prove now that this property is uniform in x: for each/3' < fl there exists ~ such that for each x ~ S there exists u such that (7) holds. By contradiction assume that there exist the sequences {xk ~ S} and rk ~ 0 such that
97
Our goal is to find a domain o f attraction i n X which is as close as possible to the largest one. We say that the C-set P internally approximates in X the C-set S with precision e > 0 if (1 - e ) S C P C X .
tPs (xk 4, zk(Axk + Bu)) - q~s (x~)
> -/~' q's (xe)
-rk
for all {u ~ U}. Since S is a compact set, the sequence {xk} admits a converging subsequence. Without restriction we assume that xk + 5. Let/~" such that -/3 < - fl" < -/~'. There exists "~and fi such that
qJs (54- £(A£ 4, Bfi)) - 7Js (5)
(9)
The next theorem, which is the main result of this section, states that we can internally approximate with arbitrary precision the largest domain o f attraction SI~ with a polyhedral domain of attraction P loosing an arbitrarily small amount in terms of speed of convergence. Theorem 2.1. Let S~ the largest domain of attrac-
~< -/3" q's(,~).
Now the difference quotient is a non-decreasing function of z if q~s is convex [13] then for zk ~<
~Ps (xk 4. zk(Axk 4. Bgt)) - 7*s (xk) ~k
tion in X for (1) with speed of convergence fl > O. Then for all 0 0 there exists a polyhedral C-set P such that (1 - e ) S ~ C P C X and such that P is a domain of attraction for (1) with speed of convergence /3. Moreover, the control can be expressed in a feedback form u = Cb(x), where 4~ is a Lipschitz.function on P.
<~ qJs (xk + i(Axk + Bfi)) - 7~s (xk) = ~s(Y:4,{(A54.Bfi))-{
7*s(~) 4.^l(5'(xk), "C
where in view of the continuity of ~s, (5'(xk) --+ 0 as xk --+ 2. So for sufficiently large k
qJs(xk 4, zk(Axk 4. B~)) - ~s(xk) "r,k
<. -/3'~s(;),
a contradiction. Therefore there exists -g such that for all x there exists u assuring (7). Note that it is not restrictive to assume that f is such that 1 - -gfl' = 2' < 1. Then
~Ps ([I 4- fA]x 4. ~Bu) <~ [1 - f/3']~s (x) = 2'Ws (x).
(8) This immediately implies that S is 2'-contractive for the EAS. Lemma 2.2. Assume that the C-set P is ),'-contrac-
tive for the E A S for some 2' such that 0 <~;t' < 1. Then P is a domain of attraction for (1) with /3' = (1 - )o')/z. Moreover, there exists a Lipschitz feedback control function q): P ~ U assuring condition (4). Proof. See [2] or [4].
[]
Proof. Take/~ < / 3 ' < fi and fix e. > 0. From Lemma 2.1 we can find z such that S/~ is contractive for the EAS with )~' = 1 - ~/3' < 1. Consider the largest )~'-contractive set R;, included in X, then S/~ C R;,. Let 2 = 1 - z f l > 2 ' , from [3], Theorem 3.2, R~, can be arbitrarily closely approximated by a polyhedral )w-contractive set P C X in the sense that (1 - e ) R ; , C P C R ; , . Now S~CR~, implies (1 - e)S/~C(1 - e.)R;, C P C X . From Lemma 2.2, the polyhedron P is a domain of attraction with speed/3 for the continuous system. The existence of a (Lipschitz) feedback control follows from [2, 4]. [] From the theorem above we have that to approximate the maximal domain of attraction with a polyhedral domain o f attraction, we have just to use the EAS with a sufficiently small ~ and to apply the method proposed in [3] to approximate the largest 2 contractive set for the EAS. The derived polyhedron is a domain of attraction for ( 1 ). Moreover, note that a set is 2-contractive for the EAS if and only if it is invariant for the following modified Euler approximating systems
x(t + 1) = [I + ZA]x(t ) Jr- =-u(t). zB A
(10)
A
This means that we can also apply the procedures proposed in [8, 10] for discrete-time system.
98
[( Blanchini, S. Miani/Systems & Control Letters 28 (1996) 95 102 1
0.8 0.6 0.4 0.2
o -0.2 -0.4 -0.6
-0.8
"'-1
-0.8
-0.6
-01,4
-0'.2
¢2
0'4
0~.6
0.8
Fig. 1. EAS (dotted) and exponential (dashed) approximation of PcoN.
Remark2.1. An important question is whether we can determine an approximation of the largest domain of attraction via exponential approximation. The answer is affirmative with a fundamental difference: the derived approximating set is not in general a
the continuous time system but it is invariant as well.
3. Feedback control
domain o f attraction f o r the continuous-time system.
In Fig. 1, we show the largest invariant set PCON C X, x -- {x: [x,I ~< 1}, u -- {u: lul ~< 1} for a double integrator system with 1
(in this simple case PCON is very easily derived). In Fig. 1 we show PEAS C X , the (internal) approximation obtained through the EAS and the polyhedral approximation obtained with the exponential approximation PEXP C X using z = T = 0.5. Since PEXP ~ PcoN, PEXP is not invariant (this is confirmed by checking vertex conditions). This limit of the exponential approximation was also mentioned in [15] where the problem of the construction of the largest invariant set contained in X for a stable linear system was considered. As a particular case with B = 0 and A stable, we solve that problem by providing a set which not only converges to the largest domain of attraction for
We assume now that a polyhedral domain of attraction P is known and we consider the problem of associating to P a feedback control. A suitable stabilizing control is the linear variable-structure one proposed in [3, 8] as follows. We associate to each vertex x of P, a control u E U such that (6) holds. Each of these control vectors "pushes" the vertex state inside the polytope P. Then we divide the polytope P into simplicial sectors Pi, each of them formed by the origin and n distinct vertices of P. According to [3, 8], the partition in sectors must be such that/oh has a nonempty interior, Ph n Pk has empty interior if k ¢: h, and [,-JkPk = P. Techniques to obtain this partition are discussed in [12]. In the sector h the linear gain u = Kh x, with Kh = Uh X~- i
has to be applied, where Xh is the matrix formed by the n non-zero vertices of Ph and Uh is the matrix of the corresponding control vectors. In [4] it is shown that this control is Lipschitz and assures (6) on P.
F, Blanch&i, X Miani/Systems & Control Letters 28 (1996) 95-102
The considered linear variable-structure control is easy to apply as long as the system dimension is small. If this is not the case, computational problems may arise due to the number o f simplicial sectors Ph which can be very high. We propose, for single input systems, a new compensator which is discontinuous. The advantage of this compensator is twofold: first it is less complex than the previous, second it is amenable for systems with quantized control since it takes values on the boundary of U. Assume that the C-set P has the following plane representation:
P = {xE~n: Fix <~ 1, i = 1..... an} = {xE ~n: Fx<<.T}.
( l 2)
Define now I(x) as the set of all indices where the maximum in (12) is reached (clearly this set is a singleton i f x is in the interior of a sector)
l(x) = {i: i = argmaxj Fix}.
The Lyapunov derivative for this function is given by
D+(x, u) = lim sup
qJp (x + ~(Ax + Bu)) - qJp (x)
r--+0
= max Fk(Ax +Bu). kEl(x)
(15)
By assumption for each x E P there exists u ~ U such that D+ (x, u) <~ - fl~p (x ). Now, to enforce maximal contractivity we can try to find a controller u = q~(x) by solving the optimization problem of minimizing D+(x,u) with respect to u. For all x in the interior int{Sk} of a sector Sk we have I(x) = {k}, then the solution of the minimization problem inf Fk(Ax + Bu) = FkAx ÷ inf (FkBu)
(11)
The polytope P can be divided in "natural" sectors Si which are the pyramids each o f them obtained as the convex hull of the origin and one facet (i.e. an (n - 1 )dimensional face) of the polytope. Given x E P, the sector including x is that associated to the ith facet, where i = argmaxj F i x.
99
(13)
Note that for a general polyhedral C-set P, the natural sectors are different from the simplicial sectors. For instance in a three-dimensional cube each square facet generates one natural sector and two simplicial sectors (which are tetrahedra). The number as of simplicial sectors is clearly greater or equal to the number an of natural sectors (the polytopes having the property an = as are called simplicial) and the ratio between the two numbers can grow exponentially with the space dimension. The main idea of the new compensator is to try to maximize the speed of convergence to the origin measured with respect to the norm induced by P. As a result, we will derive a compensator which is constant in each natural sector. Henceforth, we assume that q = 1 and that the control is constrained as u c U = [u-, u+].
uEU
uCU
is reached on the same extremum of U. Denote by u k E { u - , u +} such a solution, then the following variable-structure (bang-bang) control can be defined u = ~ B ( x ) = Uk for x r S k ,
(16)
which is constant in the interior of each sector. Since the definition of the control ~b~ is ambiguous in the intersection of several sectors associated to different controls, ambiguity is removed by arbitrarily assigning to any x which belongs to such an intersection one of the extreme values of U. It is obvious that as long as the trajectory of the system with this control evolves in the interior of the sector the condition (4) is assured. However this control may not be continuous on the inter-sector boundary and it may introduce sliding trajectories. Nevertheless, this controller assures global convergence according to the following theorem.
Theorem 3.1. Assume that the polyhedral C-set P & a domain of attraction for (1) with speed of convergence ft. Then the control q~BBdefined as (16) assures convergence with a speed fl Jor all initial conditions inP. Proof. To prove the theorem it is sufficient to prove that u = 4~BB(X) assures (6) for all x in P. The condition (6) is obviously assured i f x is in the interior of a sector. Thus, we assume that x is on an inter-sector boundary ~ individuated by Fkx = Up(x), k E I~. Define as Z the smallest subspace including ~ . Z is defined by the equalities
If P is represented as in (11 ), then the induced norm is (see for instance [16])
Fix = Fjx,
q~e(X) = max Fix.
Without restriction assume that the control is not continuous on .~ (say it takes different values on the
i--1,...,an
(14)
i,j E I~.
F Blanchini, S. Miani/Svstems & Control Letters" 28 (1996) 95 102
100
sectors incurring in ~ ) . If the state leaves .~ then the decreasing condition (6) is fulfilled and thus let us consider the case in which the motion lies on ~ for t in some nonzero interval [t0, tl). Invoking the equivalent control method [17], there must exist U~q such that u - ~< ~/eq ~ u+ and that Ax + Bueq is tangent to 27. In our case, being 2; a linear manifold, this is equivalent to Ax 4- Bueq C ~ which implies that F j ( A x 4- Bueq ) =
for some positive ft. In this case, as the state approaches the origin, we can reduce the control effort by jumping between intermediate points, say uE {-(k/r)a,(k/r)g},
k E {0, 1. . . . . r}
instead of between -~7 and ~7. The value of k can be chosen in such a way the condition D+(x,u)<~ - fTJ(x) is assured for [u[ ~< (k/r)~. By linearity it is immediate to show that, denoting by Ikt] the smallest integer greater or equal to It E R, k can be chosen as
Fi(Ax + BUeq ) f o r i,j E 1~. k(x) = FrqJ(x)].
To prove the theorem, we need to show that the equivalent control assures (6). By contradiction assume that this is not the case, then for t E [to, tl ), D*(x, Ueq ) • -- [~tlJp(X). From (15) D + ( x , Ueq) = F j ( A x 4 - B u e q ) > - [;]Up(x)
for j E / e .
(17)
By construction of P for every x there exists u* such that D+(x,u *) = max F/(Ax + Bu*) <~ - fillip(X). /
(18)
Now one of the two inequalities u*~< Ueq ~< u +, /2-- ~ Ueq ~ b/* holds. Assume u* ~< Ueq ~ U+ (the other case is obviously equivalent). For every j E f~, by the linearity of Fj(Ax +Bu), from (1 7) and ( 18) we get Fj(Ax + Bu +) > - f ~ p ( X ) for all j E/~e. Hence for all concurring sectors Sj, j E I~, the minimizing control 4~nB(X) must be equal to u - in contradiction with the assumption that the control is discontinuous on/~.
[]
As a final comment, we remark that in practical applications, the control passes through a sampling device. So, imposing to the control to take values on the extrema may cause considerable inter-sampling ripples which may be unacceptable when the state approaches the origin. This problem can be overcome by reducing the sampling time. Sampling-time reduction can be avoided if the quantized actuator can take values not only at the extrema but in a finite number of points. Assume for simplicity that U is no more an interval but a set of the form
R e m a r k 3.1. To eliminate ripples we can proceed in a different way by "switching" to a linear compensator u - K x when the state reaches a sufficiently small neighbourhood V of the origin (clearly this method does not work with quantized control). Our techniques, applied as particular case can be used to approximate the largest invariant set in V for the closed-loop system 2 = (A + BK)x by an invariant polyhedral set V*. Thus "switching" can be safely done as soon as the condition x(t) E V* is verified. Note that in this way we can improve the asymptotic behavior of the system by properly assigning the eigenvalues o f A + BK. R e m a r k 3.2. The definition of the control q~B8 can be extended to the case of multi-input systems. In this case the control assumes values on a vertex of U in each sector, but it can be easily shown that nonconvergent sliding trajectories may arise in the intersector boundary. R e m a r k 3.3. The sampling time T used to practically implement the control, has not to be equal to the parameter r we use to compute the set. In practice we should have T << r. 3.1. Robustness o f the compensator This paper together with the previous references [2-4] furnishes all the material to provide a design method which not only takes into account state and control constraints, but derives a control which is robust against structured parametric perturbations. Assume that A and B are uncertain matrices of the form r
A(w) = ~ A i w i , i--1 '
U = {-ti,
(r-1)£,r
! ~ 7 , 0 , ! f f . . . . . ~7}
B(w) = E B i w i , i 1
wi • i=1
1,
w i • O.
(19)
F. Blanchini, S. Miani/ Systems & Control Letters 28 (1996) 95-102
101
Tablel
~]
~2
1 2
1.0000e + 0 4.3805e -
0.0000e + 0
0.0000e + 0
12
1.0562e + 0
1.1072e + 1
1.6853e -- 0
3 4 5
6.9169e 6.9806e 1.2750e Y1260e 7.4470e 5.0536e 6.3672e
2.4989e 2.3446e 1.9799e 2.1248e 2.6163e 2.5717e 2.8315e 2.6869e 2.3492e
13 14 15 16 17 18 19 20 21
2.1837e 6.8219e 4.1632e 1.0444e 6.0438e 6.5936e 5.7590e 1.0204e 8.5464e
6.0240e 1.2939e 9.6618e 1.0573e 8.7796e 1.2310e 1.2281e 1.0063e 1.0610e
2.4156e -- 0 1.6666e + 0
6 7 8 9
7.3046e 1.3776e 1.3515e 4.2319e 7.7821e 9.891 le 1.1183e 1.3210e 1.1186e
k
10 11
-
9.4652e 1.0359e + 0
~3
+ + + ÷ + + + + +
0 1 1 0 0 0 I 1 1
1.1398e + 1
k
+ + -+ + + ÷ ÷ +
0 0 0 0 0 0 0 0 0
~l
2.0107e + 0
~2
--+ + -
I 1 1 0 1 1 1 0 1
--
~3
+ + ÷ + + + + + +
0 1 0 1 0 1 1 1 1
2.7826e ÷ 0 1.4167e + 0 2.5864e + 0 1.4145e + 0 2.7646e + 0 1.2055e + 0 2.4594e + 0
--
0.4-, 0.3~ 0.2~ 0.1
i :
i .
-::.5.
i "
"
"-. " . .........
.....
i ..
i... ....
-
0
.....
.
0
......
4
". "
...............
•
......
: i
..
..
.......
~
o.2
-0.2
-0.06 -0.8 Fig. 2. The simulated trajectory with sampling time T = 10 - 3 .
As shown in [3] we can form a polyhedral 2contractive polyhedral set for the EAS which is a domain of attraction for the continuous-time system [4]. As long as the control is scalar, the proposed bang-bang compensator can easily be extended by taking in the sector Sk the solution ~BB(x) -- uk of the following problem: min
u- <~u.<.u+
m a x F k ( A i x + Biu).
Proving the state convergence for the closed-loop system is a simple extension of Theorem 3.1, and we skip the proof for brevity.
4. Example Consider the following continuous-time 3-dimensional model of an armature-controlled DC electric motor. We assume that Xl is the armature current, x2 is the angular speed and x3 is the angle. The control u is the armature voltage. This is a typical example of a system in which the control quantization is an important aspect. Indeed voltage suppliers are very often able to provide values only on a finite set. So assume that the control can take values in the finite normalized set { - 1, -0.5, 0, 0.5, 1}. The state values are normalized with respect to the maximal allowable
102
F Blanchini, S. Miani/ Svstems & Control Letters 28 (1996) 95 -102
values so that X = {x E N~: A and B are
A =
-0.0700 0.0600
-0.8600 0.0085
o.oooo
Ixil ~ 1 }.
The matrices
0.0000] / 0.0000[
.oooo o.ooooj
F l.OOOOl B = /°°°0°/
LO.OOOO_l A p p l y i n g the procedure in [3] with r : 1 and ,:. = 0.8 we derived the polyhedral set P = {x" - 1 ~< F i x ~< 1, i = 1,2 . . . . . 21} w h i c h is 0 - s y m m e t r i c since X and U are both symmetric. The rows o f the matrix F are reported in Table 1. A c c o r d i n g l y to L e m m a 2.2 we have that c o n v e r g e n c e is assured with [3 = 0.2. Since the set S is symmetric each sector is associated to its opposite one. Using the linear variable structure controller we have to consider 124 simplicial sectors in each o f w h o m we apply a linear control. Using the control tbBB we have just to consider 42 natural sectors. W e denote as positive sector the one associated to the facet F k x 1 and as negative, the one associated to the facet F k x = - 1 . Since Fk can be exchanged with - F k , in Table 1 we report Fk in such a w a y that the corresponding positive sector is associated to u = - 1 ( o f course the negative sector is associated to u = +1 ). Finally, in Fig. 2 we report the 3-D simulated state trajectory from the initial point x ( 0 ) = [0 0 0.3532] r, which has been obtained by using the quantized b a n g bang control with a sampling time T = 10 -3 for an interval o f 24 s (the angle 0.3532 corresponds to the state with m a x i m u m angle and null initial speed and current belonging to P).
5. Conclusions In this paper it was shown h o w to approximate the m a x i m a l d o m a i n o f attraction for a linear continuoustime system with state and control constraints via Euler A p p r o x i m a t i n g System. W e p r o v e d that the approximating set is a domain o f attraction for the continuous-time system while this property is not assured by domains determined through the classical exponential approximation. This domain o f attraction can be associated to a continuous feedback control. For the single-input case we p r o v i d e d a variable structure controller which is no m o r e continuous, but
has the advantage o f requiring less computational effort than the previous one and being suitable for applications to quantized input systems.
References [1] A. Benzaouia and C. Burgat, The regulator problem for a class of linear systems with constrained control, Systems Control Lett. 10 (1988) 357-363. [2] F. Blanchini, Ultimate boundedness control for discrete-time uncertain system via set-induced Lyapunov functions, 30 ° IEEE Conj'. on Decision and Control, Brighton, United Kingdom (December t l 13, 1991) 1755-1760. [3] F. Blanchini, Ultimate boundedness control for discrete-time uncertain system via set-induced Lyapunov functions, IEEE Trans. Autom. Control AC-39 (1994) 428-433. [4] F. Blanchini, Non-quadratic Lyapunov function for robust control, A utomatica 31 ( 1995 ) 451-461. [5] F. Blanchini, F. Mesquine and S. Miani, Constrained stabilization with assigned initial condition set, Int. J. Control 62 (1995) 601-61% [6] C. Burgat and S. Tarbouriech, Positively invariant sets for constrained continuous-time systems with cone properties, IEEE Trans. Autom. Control 39 (1994). [7] E.B. Castelan and J.C. Hennet, Eigenstructure assignment for state constrained linear continuous time systems, A utomatica 28 (1992) 605-611. [8] P.O. Gutman and M. Cwikel, Admissible sets and feedback control for discrete-time linear systems with bounded control and states, IEEE Trans. Autom. Control AC-31 (1986) 373 376. [9] P.O. Gutman and P. Hagander, A new design of constrained controllers for linear systems, IEEE Trans. Autom. Control AC-30 (1985) 22-33. [10] S.S. Keerthi and E.G Gilbert, Computation of minimum-time feedback control laws for discrete-time systems with state and control constraints, IEEE Trans. Autom. Control AC-32 (1987)432 435. [11] J.B. Lasserre, Reachable, controllable sets and stabilizing control of constrained systems, Automatica 29 (1993) 531 -536. [12] S. Miani, Tecniche di sintesi del controllo per sistemi con vincoli su statie ingressi, Thesis, Universit~idi Padova, 1993. [13] R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1972). [14] M. Sznaier and M.J. Damborg, Heuristically enhanced feedback control of constrained discrete-time systems, Automatica 26 (1990) 521-532. [15] K. Tan and E. Gilbert, Linear systems with state and control constraints: the theory and the applications of the maximal output admissible sets, IEEE Trans. Autom. Control 36 ( 1991 ) 1008-1020. [16] M. Vassilaki and G. Bitsoris, Constrained regulation of linear continuous-time dynamical systems, Systems Control Lett. 13 (1989) 247--252. [17] V.I. Utkin, Slidiny Modes and Their Application in Variablestructure Systems (M.I.R., Moskow, 1978).