System Types in Feedback Control with Saturating Actuators.pdf

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System Types in Feedback Control with

Saturating Actuators

Yongsoon Eun, Pierre T. Kabamba, and Semyon M. Meerkov

Department of Electrical Engineering and Computer Science,University of Michigan, Ann Arbor, MI 48109-2140, USA

May 16, 2003

Abstract

This Technical Note extends the classical notion of system type to feedbackcontrol with saturating actuators. For step, ramp, and parabolic inputs, it defines

the so-called trackable domains and evaluates steady state errors. It shows that,unlike the linear case, the role of the poles at the origin of the plant and thecontroller are different and, on this basis, extends the notion of system type.Results obtained are useful for selecting controllers and actuators to ensure desiredtrackable domains and steady state errors.

This work has been supported by NSF Grant No. CMS-0073302Please address correspondence to Professor P. T. Kabamba, Department of Aerospace Engineering, Univer-

sity of Michigan, Ann Arbor, MI 48109-2140, USA, e-mail: [email protected], phone: (734) 763-6728,fax: (734) 763-0578.


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1. INTRODUCTION

The notion of system type is important in linear control systems analysis and design. This

notion, however, is not applicable to systems with saturating actuators. Indeed, the two

systems shown in Figure 1.1 have identical type if the saturation is ignored, but in the presence

of saturation have qualitatively different tracking capabilities. Namely, the system of Figure

1.1(a) can track ramps with a finite steady state error, but that of Figure 1.1(b) cannot track

any ramp. This indicates that the usual definition of system type has to be modified to be

applicable to systems with saturating actuators. This modification is the purpose of this

Technical Note.

Specifically, we show that the roles of poles at the origin of the plant and the controller are

different, and system type is defined by the plant poles. The controller poles, however, also

play a role but it is limited to affecting the steady state error, while not enlarging the class of

trackable references. This class is characterized through the new notion of Trackable Domain,

which quantifies the size of steps or the slope of ramps, that can be tracked when saturation

is present. The new system types and the Trackable Domains are the main contributions of

this work.

Systems with saturating actuators have been studied in numerous publications (see [1] for

an annotated bibliography and the recent monographs [2][4]). However, no results on system

types and Trackable Domains have been reported.

The outline of this note is as follows: Section 2 investigates the existence of steady states

and evaluates the steady state tracking errors. Based on this investigation, in Section 3,

the definition of system type is introduced and the steady state errors are characterized. In

Section 4, a design example is given. Conclusions are given in Section 5. The proofs are

included in the Appendix.

-

+r

(b)(a)

-

+r1

1

s

1

s(s+ 1)

1

s+ 1

y y

plantcontroller actuatorsaturating

controller plantactuatorsaturating

Figure 1.1: Motivating example

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2. EXISTENCE OF STEADY STATES, TRACKABLE DOMAINS,

AND STEADY STATE ERRORS

2.1 Assumption

Consider the SISO systems shown in Figure 2.1 where P(s) and C(s) are the plant and

controller, respectively, and sat(v) is the saturating actuator defined by

sat(v) =

if < v,

v if v ,

ifv 0, Q 0, (2.3)

wherex = [xTp xTc]

T.

In some cases, this assumption can be verified using the methods of absolute stability

theory [5]. In others, the method of [6] can be used. Although (2.3) seems somewhat exacting,

-+ v y

P(s)er

C(s) sat(v)

Figure 2.1: Feedback control system with saturating actuator

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we need it to ensure the stability of (2.2) with not only the symmetric saturation (2.1) but

also with every asymmetric saturation defined by sat, (v) = sat(v+) , with ||< .

Systems with asymmetric nonlinearities arise in the analysis of (2.2) when r(t)= 0.

2.2 Step Input

Consider the closed loop system (2.1), (2.2) and assume that r(t) =r0u(t), where u(t) is the

unit step function. For s R, define

C0= lims0+

C(s), (2.4)

P0 = lims0+

P(s), (2.5)

where the limits (2.4) and (2.5) are allowed to be infinite. Introduce the steady state error

with respect to stepestepss = lim

t[r0u(t) y(t)]. (2.6)

Theorem 2.1. Under Assumption 2.1,

(i) estepss exists if

|r0|0,

(c) M0,

(d) (A+BC, M) is observable.

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Proof: Using (A.1), rewrite (2.2) with r = 0 as

x= Ax+B sat(v),

v= C x.(A.3)

Under Assumption 2.1, global asymptotic stability of closed loop system (2.1), (2.2), or equiv-

alently the closed loop system (2.1), (A.3), can be established using the Lyapunov function

V(x) =xTQx+

v0

sat()d. (A.4)

If|Cx| , this V(x) reduces to

V(x) =xT(Q+1

2CTC)x, (A.5)

the derivative ofV(x) along the trajectory of (2.1), (A.3) becomes

V(x) =x

T

Mx, (A.6)

and (A.3) can be written as

x= (A+BC)x. (A.7)

Since the closed loop system (2.1), (A.3) is globally asymptotically stable, A+BC in (A.7)

is Hurwitz, which proves (a). SinceV(x)> 0 and V(x) 0, matrices Q + 12CTCand M (see

(A.5) and (A.6)) are positive definite, and negative semidefinite, respectively. This proves (b)

and (c). Finally, (a), (b), (c) imply that the pair (A + BC, M) is observable [9], which proves

(d).

Lemma A.2. Let Assumption 2.1 hold. Then, for every asymmetric saturation, sat, (v) =

sat(v+) , with ||< , the closed loop system (2.2) with r = 0, i.e.,

xp= Apxp+Bp sat, (v),

xc = Acxc+Bc(y),

y= Cpxp,

v= Ccxc+Dc(y),

(A.8)

is globally asymptotically stable.

Proof: Under Assumption 2.1, global asymptotic stability of closed loop system (2.1), (2.2)

can be established using the Lyapunov function

V(x) =xTQx+

v0

sat()d, (A.9)

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wherex= [xTp xTc]

T. It will be shown that, based on this V(x), a Lyapunov function can be

found that establishes global asymptotic stability of (A.8).

Without loss of generality, assume 0 < < . Introducing the state transformation

=

+xand the notation =

+v, rewrite (A.8) as

= A+B (),

= C .(A.10)

Here A, B , and Care defined in (A.1) and () =

+sat,

+

, i.e.,

() =

, if < ,

, if < < ,

, if 0. This must satisfy(C)= sat(C), i.e.,C > , otherwise, it would result

in V(x)>0 at x= . Define x =

. Then, Cx > , (C) = and sat(Cx

) = .

Substitutingx in (A.14) yields

V(x

) =xT

(AT

Q+QA)x

+ sat(Cx

)(2BT

Q+CA)x

+CB[sat(Cx

)]2

= (/)2T(ATQ+QA) +(2BTQ+CA)/+CB2

= (/)2[T(ATQ+QA) +(2BTQ+CA) +CB2]

= (/)2[T(ATQ+QA) +(C)(2BTQ+CA) +CB[(C)]2]

= (/)2V1()> 0,

(A.15)

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which contradicts Assumption 2.1. Therefore, V1() 0 for all = 0.

Next we show, again by contradiction, that the only solution of (A.10), (A.11) that is

contained in { |V1() = 0} is the trivial solution (t) 0.

Let (t), t 0, be a non-trivial solution of (A.10), (A.11) that satisfies V1((t)) 0.

Assume firstC(t) for all t 0. Since sat

(C(t)) =(C(t)), system (2.1), (2.2) and

system (A.10), (A.11) are identical; therefore, x(t) = (t) is a non-trivial solution of (2.1),

(2.2) as well. Moreover, as it follows from (A.13) and (A.14), V(x(t)) = V1((t)) 0. This

contradicts Assumption 2.1. Hence, (t) cannot satisfy C(t) for all t 0.

Assume now that C(t) for all t 0. Then, from (A.10), (A.11),

(t) =eAt(0) +

t0

eABd. (A.16)

Define x(t) = (t). Then, x(t) is a solution of (2.1), (2.2), since Cx(t)> and

x(t) =

(t) =eAtx(0) +

t0

eABd. (A.17)

Moreover, using the chain of equalities similar to (A.15), we obtain V(x(t)) = (/)2 V1((t))

0. This again contradicts Assumption 2.1. Hence, (t) cannot satisfy C(t) for all t 0

either. Therefore, there must exist an interval (t1, t2), t1 = t2, such that |C(t)|< for all

t (t1, t2). In this interval,

(t) =e(A+BC)(tt1)(t1), t (t1, t2), (A.18)

and

V1((t)) =T(t)M(t) = 0, t (t1, t2), (A.19)

whereM is defined in (A.2). Since Mis negative semidefinite by Lemma A.1, it follows that

M(t) =M e(A+BC)(tt1)(t1) = 0, t (t1, t2). (A.20)

This, however, contradicts the observability of (A +BC,M), which must take place according

to Lemma A.1. Thus, (t) 0 is the only solution of (A.10), (A.11) that is contained in

{ |V1() = 0}.

Finally, it can be shown, again by contradiction, that V1() > 0 for all = 0 and that

V1() as|| . Therefore, the system (A.10) and (A.11), and, hence, (A.8) is globally

asymptotically stable.

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Proof of Theorem 2.1: Three cases are possible: (a)|C0| = and |P0| =; (b)|C0|=

and|P0| =; (c)|P0|= .

(a) Assume |C0| = and |P0| =. Define xp, xc , y

, and v as

v = C0

1 +C0P0r0, y

= C0P01 +C0P0

r0, x

p = A1

p Bp v, xc =A

1c Bc(r0 y

). (A.21)

Using

xp= xp x

p, xc = xc xc , y= y y

, v= v v, (A.22)

rewrite (2.2) as

xp= Apxp+Bp sat, v(v),

xc = Acxc+Bc(y),

y= Cpxp,

v= Ccxc+Dc(y).

(A.23)

Note that condition (2.7) implies|v|< . Therefore, Assumption 2.1 and Lemma A.2 ensure

the global asymptotic stablility of (A.23). Hence, the steady state exists and y(t) converges

toy, i.e., estepss exists. This proves (i). Moreover, by definition, estepss is given by

estepss =r0 y =r0

C0P01 +C0P0

r0= r0

1 +C0P0, (A.24)

which proves (ii).

(b) |C0|= implies that Ac has an eigenvalue 0. Choose xc = 0 to satisfy

Acxc = 0, Ccx

c =

r0P0

, (A.25)

and let

v = r0P0

, y =r0, x

p= A1

p Bpv. (A.26)

Proceeding similarly to (a), it can be shown that if|r0|< |P0|, then estepss = 0.

(c) |P0|= implies Ap has an eigenvalue 0. Choose x

p= 0 to satisfy

Apx

p= 0, Cpx

p= r0, (A.27)

and let v = 0, y = r0, xc = 0. Similarly to (a), it can be shown that e

stepss = 0. This

completes the proof.

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-+ v(t) y(t)sP(s)e(t)

(b)

1s C(s) sat(v)

r1

us

(t)

-+ v(t) y(t)e(t)

(a)

P(s)1

s C(s) sat(v)

r1us(t)

Figure 5.4: Block diagram of a system with ramp input

Proof of Theorem 2.2: Consider the system with ramp input shown in Figure 5.4(a).

Equivalently, it can be represented as shown in Figure 5.4(b) where the input and output are

step and y rather than ramp and y as in Figure 5.4(a). We refer to these systems as system(a) and (b), respectively. Since the input to system (b) is step, erampss of system (a) can be

analyzed using Theorem 2.1 applied to system (b). Indeed, Assumption 2.1 holds for system

(b) since it is assumed to hold for system (a). Also, introducing

P(s) =sP(s), C(s) =1s

C(s), (A.28)

and noting that

P0= P1,

C0= , (A.29)

condition (2.17) for system (a) is equivalent to |r1| < 1C0 +P0

for system (b). Thus,Theorem 2.1 ensures the existence of steady state in system (b) and

limt

e(t) = r1

1 +P0C0 = 0. (A.30)This implies that limt e(t) = const., which proves (i).

It follows from the proof of Theorem 2.1 that | limt v(t)| < . Therefore,erampss must

be identical to that of the system with linear actuator, i.e.,

erampss = r1

limt sP(s)C(s). (A.31)

SinceP1= limt sP(s)= 0, it follows that

limt

sP(s)C(s) = limt

sP(s) limt

C(s) =P1C0, (A.32)

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and

erampss = r1P1C0

. (A.33)

This proves (ii).

The proof for the parabolic input case is similar.

REFERENCES

[1] D. S. Bernstein, and A. N. Michel, Chronological bibliography on saturating actuators,

Int. J. Robust and Nonlinear Contr., vol. 5, no. 5, pp. 375-380, 1995.

[2] A. Saberi, A. A. Stoorvogel, and P. Sannuti, Control of Linear Systems with Regulation

and Input Constraints, London: Springer, 2000.

[3] T. Hu, Z. Lin, Control Systems with Actuator Saturation : Analysis and Design, Boston:

Birkhaser, 2001.

[4] V. Kapila (editor),Actuator Saturation Control, New York: Marcel Dekker, 2002.

[5] K. S. Narendra and J. H. Taylor,Frequency Domain Criteria for Absolute Stability, New

York: Academic Press, 1973.

[6] F. Tyan and D. S. Bernstein, Global stabilization of systems containing a double inte-

grator using a saturated linear controller, Int. J. Robust and Nonlinear Contr., vol. 9,

pp. 11431156, 1999.

[7] H. J. Sussmann and Y. Yang, On the stabilizability of multiple integrators by means of

bounded feedback controls, Proc. Conf. Dec. Contr., pp. 7072, Brighton, U.K., 1991.

[8] J. Goncalves, Quadratic surface Lyapunov functions in global stability analysis of sat-

uration systems, Proc. Amer. Contr. Conf., Arlington, VA, June 2001.

[9] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, New Jersey: Prentice

Hall, 1996.

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