optimal nonlinear, but continuous, feedback control of systems...

INT. J. CONTROL, 1995, VOL. 62, No.5, 1209-1216

Optimal nonlinear, but continuous, feedback control of systems withsaturating actuators

DENNIS S. BERNSTEINt

This paper presents a modification of the optimal saturating feedback controllaws given by Frankena and Sivan (1979) and Ryan (1982a) for asymptoticallystable systems. Unlike their results, which involve bang-bang action andsingular extremals, the modified control law is continuous. Specifically, the newcontrol law is linear inside a cylinder set and saturated elsewhere. Usingsteady-state Hamilton-Jacobi-Bellman theory, this control law is shown to beoptimal for a modified performance functional with a discontinuous integrand.

1. Introduction

Since all real actuators are subject to physical constraints, one of the mostwidespread and fundamental problems in control engineering is coping withactuator saturation. The literature on this problem is extensive and reflectsconsiderable current activity, e.g. Keerthi and Gilbert (1987), Astrom andRundqwist (1989), Campo et al. (1989), Sontag and Sussmann (1990), Sznaierand Damborg (1990), Dolphus and Schmitendorf (1991), Sussmann and Yang(1991), Teel (1992), Tsirukis and Morari (1992), and Yang et al. (1992). Thepurpose of this paper is to approach this problem within an optimal nonlinearfeedback control framework as in Frankena and Sivan (1979), Ryan (1982a, b)but with certain crucial differences described below.

The starting point for our approach is Frankena and Sivan (1979) whichconsiders the linear plant

i = Ax + Bu, x(O) = Xu

with an ellipsoidal control constraint set

Q ~ {u E IRm: llTRzu ",;; I}

(1)

(2)

where Rz > O. If the control were not confined to Q, then one could apply theusual linear-quadratic approach which involves minimizing the performancefunctional

J = f'ixTR1X + llTRzll]dt (3)

where Rl ~ O. Assuming that (Rt> A) is detectable and (A, B) is stabilizable,minimizing J yields the linear feedback control law u = cp(x),where

cp(x) = -R:;I BT Px (4)

Received 25 March 1993. Revised 9 September 1993.t Department of Aerospace Engineering, The University of Michigan, Ann Arbor. MI

48109-2140, U.S.A. Tel: +(1) 313 764 3719: Fax: +(1) 313 763 0578: e-mail: [email protected].

0020.7179195 $10.00 If) IW~ Tilylor tV Fronr;, I In

1210 D. S. Bernstein

where P ~ 0 satisfies the Riccati equation

0= AT P + PA + Rt - PSP (5)

where S ~ BR:;t BT. Although the feedback control law (4) yields a globallyasymptoticallystable closed-loopsystem, u = CP(x)can clearly violate the controlconstraint given by Q.

To account for the control constraint set Q, we confine out attention, as inFrankena and Sivan (1979) and Ryan (1982a), to open-loop asymptoticallystable systems. In this case, the approach in Frankena and Sivan (1979) involvesthe modified performance functional

00

J = Jo [x TR1x + 2(x TPoSPox) 1/2]dt (6)

where Po is the solution to the Lyapunov equationo = AT Po + PoA + Rl (7)

Note that Po exists and is non-negative definite since A is assumed to beasymptotically stable. Although the term 2(x TPoSPox)1/2in (6) is not quadratic,it serves to penalize further the deviation of the state x from the origin.Furthermore, although the presence of Po in the performance functional (6) issomewhat contrived, this approach is completely consistent with the techniquesused by Bass and Webber (1966), Speyer (1976) and Bernstein (1993). Specific-ally, the nonlinear control laws obtained in these papers were based uponnon-quadratic performance functionals involving auxiliary terms whose presencewas used to obtain closed-form solutions to the steady-state Hamilton-Jacobi-Bellman equation. In the case of (6), the auxiliary term 2(xTPoSPoX)1/2appears in place of the usual quadratic control-weighting term uTR2u.

As shown in Frankena and Sivan (1979), the optimal feedback control lawu = CP(x) for (6) is given by

CP(x)= (xTPOSPoX)-l/2R:;lBTpox (8)

if BTPox =1=O. If, however, BTPox = 0, then (8) must be replaced by a singularcontrol law which requires special analysis to determine whether the controlconstraint is satisfied. Nevertheless, in the non-singular case, that is, BTPox =1=0,the control law (8) is bang-bang and is thus discontinuous. The purpose of thispaper is to replace (8) with a continuous control law that is guaranteed to satisfythe control constraints. As will be seen, an additional advantage of this modifiedcontroller is that special treatment of the singular control is no longer needed.In addition to the ellipsoidal constraint set (2) we obtain analogous results forthe rectangular control constraint set considered by Ryan (1982a).

2. Continuous saturating controlsWe now replace the performance functional (6) used by Frankena and Sivan

(1979) with the modified performance functional00

J= JO[xTR1X+ h(x,u)]dt (9)where

XTPoSPox < 1x TPoSPox ~ 1

(10)

Feedback control of systems with saturating actuators 1211

and where Po is given by (7), and we now assume Rl > O. Although this minormodification of (6) leaves the optimal feedback control (8) unchanged forx TPoSPox ;;a:1, the control law for x TPoSPox< 1 (and, in particular, forBTPox = 0 in which case x TPoSPox= 0) is now quite different. Specifically.applying Theorem A.l of the Appendix yields the optimal stabilizing feedbackcontrol law

{

-I T T_ -R2 B Pox, x PoSPox < 1

cf>(x)- _(xTpOSPox)-l/2RzlBTpox, xTpoSPox;;a:1 (11)

Closed-loop stability is guaranteed by the Lyapunov function Vex) = x TPoxwhose derivative is given by

. ) _{

-xT(R1 + 2PoSPo)x,vex - -xT[Rl + 2(XTPOSPOX)-1/2POSPO]x.

Details are given in the Appendix.The form of hex, u) in (10) can be viewed as a merging of (3) and (6) so

that, in the region xTpOSPox < 1, (9) yields the usual linear LQR controller (4)with P = Po, while, in the complementary region x TPoSPox;;a:1, (9) yields thesaturated control law (8). Furthermore, these regions are chosen so that at theircommon boundary the control law is continuous. Note that in order to achievethis continuous control law the function h(x, u) was chosen to be a discontinu-ous function of x and u. This is permitted by Theorem A.I of Appendix Agiven by Bernstein (1993).

In discussing (11), it is convenient to define the set

<{6 ~ {x E IRn:x TPoSPox < I}

XTpOSPOX < 1x TPoSPox ;;a:1

(12)

To see that <{6is a cylinder set with ellipsoidal cross-section, apply an orthogonalcoordinate transformation to PoSPoso that in the new basis PoSPohas the form

[~ ~Jwhere the size of the positive diagonal matrix D is equal to the rank of PoSPo.Thus, <{6corresponds to the translation of an ellipsoid along the null space ofPoSPo.

It can now be seen that cf>(x)given by (11) is a continuoussaturation-typecontrol law that remains within the control constraint set Q. Outside of thecylinder set <{6,the control cf>(x)is saturated, while inside <{6the control is linearand thus the control effort diminishes gracefully to zero. In fact, duringoperation, the system is free to pass in and out of <{6without large, discontinuouscontrol variations. Note, however, that the linear portion of cf>(x)is differentfrom the usual LQR control law (4) since Po is given by the Lyapunov equation(7) rather than the Riccati equation (5). This construction is allowable since A isassumed to be asymptotically stable. Finally, since cf>(x) is defined globally, nospecial attention is needed for the case BTPox = 0, as in Frankena and Sivan(1979).

3. Rectangular control constraint setWe now consider the rectangular control constraint set

Q = {u = [Ul U2 ... Um]T E IRm: IUil ~ ai' i = 1, . . ., m} (13)


where a; > 0, i = 1, . . ., m, in place of the ellipsoidal control constraint set (2).In most applications the rectangular control constraint set (13) provides a morerealistic model of control constraints than the ellipsoidal control constraint setsince each control signal U; (and hence the corresponding actuator) saturatesindependently of the other control signals. Optimal, discontinuous controllersfor this problem were given by Ryan (1982a, b).

Applying the same ideas as in § 2, we consider the cost functional (9) butnow with h(x, u) given by

In

hex, u) = 2:h;(x, u);=1

(14)

where, for i = 1, . . ., m,

h ( )_

{(BTPoX)2 + uT, IBTpoxl < a;

.xu - T TI , 2a;IB; poxl, IB; Poxl ;;,.a;

where B; denotes the ith column of B. Note that if IB TPoxI< a; for all i = 1,..., m, then hex, u) = [2:~lxTpoB;BTpox] + uTu = xTpBBTpx + uTu, whichis identical to (10) for the case x TPoSPox< 1 with R2 equal to the identitymatrix.

With (15) we consider the feedback control law CP(x)= [CPl(X).. . CPm(x)]T,where, for i = 1, . . ., m, CP;(x)is given by

J

T-B; Pox,

CP;(x)=

l

-a;BTPox I T IB. Rox ;;,.a.T ' I I

IB; poxl

As before, this control law is linear within a cylinder set which now haspolygonal cross-section. Outside of this cylinder set at least one component ofCP(x)is saturated, while CP(x)is continuous throughout IRn.

Closed-loop stability is guaranteed by means of the Lyapunov functionVex) = x TPox whose derivative is given by

Vex) = -[XTR1X + 2.L IBTpoxl2+ 22: a;IBTPoxl

]IE Tlin IEIsa.

where Ilin(x) ~ {i: IBTpoxl < a;} and Isat(x) ~ {i: IBTpoxl;;,. a;} denote the un-saturated and saturated components of CP(x),respectively. Finally, optimality isguaranteed by condition (A 8), which is of the form

H(x, V,T(x), u) = 2: (BTpox + u;)2 + 22: [a;IBTPoxl + BTpoxu;] (18);Ehn ;ETsa,

(15)

(16)

(17)

Finally, with (16), it follows that H(x, V'T(X), CP(x»= 0 so that (A 7) holds,which verifies Theorem A.l.

4. Output feedback dynamic compensationWe now consider the more practical case in which only measurements are

available for feedback. Hence, consider

:i: = Ax + Bu (19)

y = Cx (20)

-----


with the observer-based output feedback dynamic compensator

Xc = AXe + Bu + Be(Y - Cxe) (21)

u = cp(Xc) (22)

where u E Q c IRm,y E 1R1,XcE IRnand cp(O)= O. The following result, which isbased on the approach of Yang et at. (1992), guarantees closed-loop stabilityusing the optimal saturated controllers given in §2 and 3.

Proposition 4.1: Assume that A and A - BcC are asymptotically stable matrices,and let cp(x) be given by either (11) or (16) with Po given by (7) with Rl > O.Then, the zero solution of the closed-loop system (19)-(22) is globally asymptot-ically stable.

Proof: Define e ~ Xc - X and write (19)-(22) as

X = Ax + Bcp(x + e)

e = (A - BcC)e

Now let Re be an n x n positive-definite matrix such that

Rl + 2PoSPo > POSPOR;lPoSPo

(23)

(24)

(25)

and let Pe> 0 satisfy

o = (A - BeC)TPe + Pe(A - BeC) + Re (26)

Now, by defining the positive-definite Lyapunov candidate V(x, e) by

V(x, e) = xTpox + eTPee (27)

it follows from the ellipsoidal control constraint set with control given by (11)that

V(x, e) = _[;]T[Rl ;o~~~SPo P~ePo][;], f3(x,e) < 1I

= _[X

]T

[Rl + 2f3(x, e)-1/2poSpo f3(x, e)-I/2PoSpo

] [X

]I

e f3(x, e)-1/2poSpo Re e

f3(x, e) ~ 1

(28)

where f3(x, e) ~ (x + e? PoSPo(x + e). For the rectangular control constraint seta slight modification of (28) is needed. From (25) it follows that

Rl + 2f3(x, e)-I/2PoSPo> f3(x, e)-l POSPOR;lPoSPo (29)

for both f3(x, e) < 1 and f3(x, e) ~ 1. Thus, both matrices in (28) are positivedefinite. Consequently, V(x, e) is negative definite, as required. 0

Remark 4.1: The Lyapunov function proof of Proposition 4.1 takes the place ofthe CICS (converging-input converging-state) property used by Yang et al.(1992) to prove asymptotic stability of the observer-based dynamic compensator.While our results are limited to open-loop asymptotically stable plants, theresults of Sontag and Sussmann (1990), Sussmann and Yang (1991) and Yang etal. (1991) apply to plants with poles on the imaginary axis. As shown by


Sussmann and Yang (1991), plants such as the multiple integrator are notstabilizable by means of a saturation of a linear control law, which is preciselythe form of (11) and (15). 0

Remark4.2: Although the control law given by Proposition 4.1 is globallyasymptotically stabilizing, it is not guaranteed to be optimal for some perform-ance functional. Since the open-loop system is assumed to be stable, it isnecessary to compare open and closed-loop performance in order to determinethe benefits of implementing (21), (22). 0

ACKNOWLEDGMENT

The author thanks Elmer Gilbert for helpful discussions and suggestions.This work was supported by the Air Force Office of Scientific Research undergrant F49620-92-J-0217.

AppendixFirst we quote the steady-state Hamilton-Jacobi-Bellman result from Bern-

stein (1993). Let f: IRnx Q_lRn, where 0 E Q C IRm,and consider the system

i = f(x, u), x(O) = Xo, t;;::.0 (A 1)

where f(O,O) = O. A control u(.) is admissible if it is measurable and u(t) E Q,t ;;::.O. Furthermore, let L: IRn x IRm-IR and, for p E IRn, define H(x, p, u) ~L(x, u) + pTf(x, u). The following result is proved in Bernstein (1993).

Theorem A.I: Consider the controlled system (A 1) with performance functional

J(xo, u(.» ~ LooL(x(t), u(t» dt (A 2)Assume that there exist a c1 function V: IRn_ IRand a function cp: IRn_ Q suchthat

V(O) =0

V(x) > 0, X E IRn,x =1=0

CP(O)= 0

V'(x)f(x, cp(x» <0, x E IRn, x =1=0

H(x, V'T(x), cp(x» = 0, x E IRn

H(x, V'T(x), u);;::.0, X E IRn, u E Q

(A3)

(A 4)

(A5)

(A6)

(A 7)

(A 8)

Furthermore, assume that V(x) - 00as Ilx11_ 00. Then, with the feedback controllaw u(') = CP(x(.», the solution x(t) = 0, t;;::'0, of the closed-loop system isglobally asymptotically stable, and

J(xo, CP(x(.») = V(xo) (A 9)

Finally, the feedback control u(.) = cp(x(.» minimizesJ(xo, u(.» in the sensethat

J(xo, CP(x(.))) = min J(xo, u( .»u( . )E~f(xo)

(A 10)


where

g(xo) ~ {u(.): u(.) is admissible and x(.) given by (A 1) satisfies

limt-+", V(x(t» = O}

Now consider the problem of finding admissible u( . ) to minimize (9), whereQ is given by (2). To do this let Vex) = xTpox, where Po satisfies (7), and letCP( . ) be given by (11). Then clearly cp:IRn- Q and (A 3)-(A 5) are satisfied.For the case x TPoSPox < 1, we have

V'(x)f(x, CP(x»= -x T(R! + 2PoSPo)x

which verifies (A 6). Next, we obtain H(x, V'T(x), u) = (u + R:;l BT pox)T R:;!(u + R:;l BT Pox), which proves (A 7) and (A 8). For the case x TPoSPox ;=:1with CP(x) given by (11), it follows that

V'(x)f(x, CP(x» = -x T[Rl + 2(x TPoSPox)-l/2 PoSPo]x

which implies (A 6). To obtain (A 7) note that

H(x, V,T(x), u) = 2[(x TPoSPoX)1/2 - (-X TPoBR:;1/2)(RY2U)]

;=:2(x TPoSPox)!/2[1 - (UTR2u)1/2]

;=:0

Thus, (A 8) is satisfied by (11) since cpT(x)R2CP(x)= 1.

Remark: In the case x TPoSPox ;=:1, the control law CP(x) must bechosen to satisfy x T PoBCP(x) ~ 0 and cpT(x)R2CP(x) = 1. Although CP(x)=_(xTPOSPOX)1/2R:;lBTpOX given by (11) satisfies these conditions, it is clearlynot the only choice of CP(x)to do so. However, with CP(x)= -R:;lBTpox for thecase x T PoSPox < 1, it follows that the resulting control law (11) is continuous onIRn. 0

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