System Types in Feedback Control with Saturating Actuators

Transcription

1 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 , USA May 6, 23 Abstract This Technical Note extends the claical notion of system type to feedback control 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 the controller are different and, on this basis, extends the notion of system type. Results obtained are useful for selecting controllers and actuators to ensure desired trackable domains and steady state errors. This work has been supported by NSF Grant No. CMS-7332 Please addre correspondence to Profeor P. T. Kabamba, Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI , USA, phone: (734) , fax: (734)

2 . 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. have identical type if the saturation is ignored, but in the presence of saturation have qualitatively different tracking capabilities. Namely, the system of Figure.(a) can track ramps with a finite steady state error, but that of Figure.(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 cla of trackable references. This cla 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 [] 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. saturating controller actuator plant controller saturating actuator plant r + - s(s + ) y r + - s s + y (a) (b) Figure.: Motivating example

3 2. EXISTENCE OF STEADY STATES, TRACKABLE DOMAINS, AND STEADY STATE ERRORS 2. Aumption Consider the SISO systems shown in Figure 2. where P(s) and C(s) are the plant and controller, respectively, and sat α (v) is the saturating actuator defined by α if α < v, sat α (v) = v if α v α, (2.) α if v < α. In state space form, this system can be represented as: ẋ p = A p x p + B p sat α (v), ẋ c = A c x c + B c (r y), y = C p x p, (2.2) v = C c x c + D c (r y), where x p and x c are the states of P(s) and C(s), respectively. Results derived in this note are obtained under the aumption that the closed loop system (2.), (2.2) has a sufficiently strong stability property characterized below: Aumption 2.. The closed loop system (2.), (2.2), with r =, is globally asymptotically stable and this fact can be established by means of a Lyapunov function of the form where x = [x T p x T c ] T. v V (x) = x T Qx + sat α (τ)dτ >, Q, (2.3) In some cases, this aumption 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, r + e v y C(s) sat α (v) P(s) - Figure 2.: Feedback control system with saturating actuator 2

4 we need it to ensure the stability of (2.2) with not only the symmetric saturation (2.) but also with every asymmetric saturation defined by sat α, β (v) = sat α (v + β) β, with β < α. Systems with asymmetric nonlinearities arise in the analysis of (2.2) when r(t). 2.2 Step Input Consider the closed loop system (2.), (2.2) and aume that r(t) = r u(t), where u(t) is the unit step function. For s R, define C = lim C(s), (2.4) s + P = lim P(s), (2.5) s + where the limits (2.4) and (2.5) are allowed to be infinite. Introduce the steady state error with respect to step e step Theorem 2.. Under Aumption 2., (i) e step exists if (ii) if r satisfies (2.7), then = lim t [r u(t) y(t)]. (2.6) r < + P C α; (2.7) e step = r + C P. (2.8) Remark 2.. Since Aumption 2. precludes pole-zero cancellation between P(s) and C(s) at s =, the terms C and P in (2.7) cannot be simultaneously infinite. The range of r, defined in (2.7) is called the Trackable Domain (TD). As indicated in (2.8), within TD, e step is identical to that of systems with linear actuators. Expreion (2.7) shows that the set TD becomes larger as the d.c. gain of the plant increases and becomes smaller as the d.c. gain of the controller increases. Thus, if P(s) has a pole at the origin, TD is the whole real line R and e step to a pole at the origin of C(s), e step is zero. Outside the Trackable Domain, i.e., if r /C + P, e step is zero. Otherwise, TD is finite, even if, due may or may not exist. It does not exist if the closed loop system (2.), (2.2) has no stable equilibrium. For instance, 3

5 consider the system of Figure 2. with parameters If (2.9) is realized as C(s) =, P(s) = s + s 2, α =. (2.9) + ẋ = x 2, ẋ 2 = x + sat α (r y), (2.) y = x + x 2, Aumption 2. holds with Q = I. However, if r + P C + 2x 2 () 2 + 2( x ()) 2, the solution of (2.) is given by and e step does not exist. x (t) = ( x ())cos(t) + x 2 ()sin(t), x 2 (t) = ( x ())sin(t) + x 2 ()cos(t), There seem to be no general methods for characterizing when e step Trackable Domain. However, if it does exist, then (2.) exists outside the e step = r sign(r C )P α. (2.2) In this case, the behavior of e step as a function of r is illustrated in Figure Ramp and Parabolic Inputs The situation here is similar to that described above. Indeed, consider (2.), (2.2) and aume that r(t) is either r tu(t) or 2 r 2t 2 u(t), where u(t), as before, is the unit step function. For s R, introduce and denote P = lim sp(s), (2.3) s + P 2 = lim s + s2 P(s), (2.4) e ramp = lim [ r tu(t) y(t) ], (2.5) t [ e par r2 = lim t 2 t2 u(t) y(t) ]. (2.6) 4

6 e step + P α C slope = + C P slope = r + P α C Trackable domain Figure 2.2: e step as a function of the magnitude of the step input r Theorem 2.2. Under Aumption 2., (i) e ramp and e par exist if r < P α, (2.7) r 2 < P 2 α, (2.8) respectively; (ii) if r and r 2 satisfy (2.7) and (2.8) respectively, then e ramp = e par = r, (2.9) C P r 2. (2.2) C P 2 The ranges of r and r 2, defined by (2.7) and (2.8), are again referred to as Trackable Domains. Thus, in the case of ramp and parabolic inputs, TD is independent of the controller and is empty if the plant has no pole at the origin, irrespective of the number of such poles in the controller. Remark 2.2. If C(s) has a pole at the origin, the system of Figure 2. is sometimes augmented by an anti-windup compensator. The purpose of anti-windup is to improve transient response and stability while maintaining small signal ( v(t) < α) behavior identical to that of the 5

7 closed loop system without anti-windup. Thus, if v(t) in the system with anti-windup does not saturate at steady state, its asymptotic tracking properties remain identical to that of the system without anti-windup. The condition on r in Theorem 2. (r, r 2 in Theorem 2.2, respectively) is sufficient to ensure lim t v(t) < α. Therefore, Theorems 2. and 2.2 remain valid in the presense of anti-windup as well. 3. SYSTEM TYPES Theorems 2. and 2.2 show that the roles of C(s) and P(s) in systems with saturating actuators are different. Therefore, system type cannot be defined in terms of the loop transfer function C(s)P(s), as in the linear case. To motivate a claification appropriate for the saturating case, consider the system of Figure.(b). With P(s) and C(s) indicated, system (2.), (2.2) satisfies Aumption 2.. According to Theorem 2., this system tracks steps from TD = {r R : r < α}, with zero steady state error. However, this system does not track any ramp input with finite steady state error. Thus, this system is similar to a type linear system but not exactly: it does not track ramps at all. Therefore, an intermediate system type is neceary. This observation motivates the following definition. Definition 3.. The system of Figure 2. is of type k S, where S stands for saturating, if the plant has k poles at the origin. It is of type k + S poles at the origin. if, in addition, the controller has one or more In terms of these system types, the steady state errors and trackable domains in systems with saturating actuators can be characterized as shown in Table 3.. As it follows from Definition 3., the system of Figure.(a) is of type S, whereas that of Figure.(b) is of type + S. This, together with Table 3., explains their tracking capabilities alluded to in the introduction. Remark 3.. Note that the definition of system type k S is a proper extension of types for linear systems. Indeed, when the level of saturation, α, tends to infinity, all Trackable Domains become the real line R and type k S systems become type k. Remark 3.2. Theorems 2. and 2.2 require Aumption 2., i.e., the existence of Q. In some cases, this may be difficult to verify. However, if existence and global stability of an 6

8 Table 3.: Steady state errors and Trackable Domain for various system types. r(t)=r u(t) r(t)=r tu(t) r(t)= 2 r 2t 2 u(t) Type TD e step TD e ramp TD e par S {r R: r < α} r +P C Not applicable Not applicable +P C + S {r R: r < P α} Not applicable Not applicable S R {r R: r < P α} r P C Not applicable + S R {r R: r < P α} Not applicable 2 S R R {r 2 R: r 2 < P 2 α} r 2 P 2 C equilibrium have been ascertained for given r (respectively, r and r 2 ), then the claims in Theorem 2. (respectively Theorem 2.2) still hold. Remark 3.3. In Table 3., only types up to 2 S are included since systems of higher types can only be stabilized locally by linear controllers with actuator saturation [7]. Trackable Domains and steady state errors for systems of type higher than 2 S can also be characterized if the aumption on global stability of the closed loop system (2.) and (2.2) is replaced by local stability and initial conditions are restricted to the domain of attraction. This, however, would neceitate estimation of the domain of attraction, which is generally difficult. Moreover, the Trackable Domain will depend on this estimate. Therefore we limit our analysis to globally asymptotically stable cases. 4. EXAMPLE In this section, we give an illustrative example of the controller design proce, using the new system types. Specifically, for a given plant with saturating actuator, we select a controller structure to satisfy given steady state performance specifications and then adjust the controller parameters to improve transient response. Consider a DC motor modeled by P(s) = s(js + b), (4.) where J =.6 kg m 2 and b =. kg m 2 /sec. Aume that the maximum attainable torque of the motor is 2.5 N m, i.e., α = 2.5. (4.2) 7

9 Using this motor, the problem is to design a servo system satisfying the following steady state performance specifications: (i) steps of magnitude r < rad should be tracked with zero steady state error; (ii) ramps of slope r < 2 rad/sec should be tracked with steady state error le than or equal to.r. A solution to this problem is as follows: According to Table 3., specification (i) requires a system of type at least + S. Since the plant has a pole at the origin, the system is of type at least S, which guarantees e step and = and T D = R. Specification (ii) requires P α 2, (4.3) e ramp = r P C.r. (4.4) Inequality (4.3) is satisfied, since P = and α = 2.5. Inequality (4.4) is met if C. Thus, a controller of the form C(s) = C n i= ( s/z i ) ( s/p i ) (4.5) guarantees (i) and (ii), if C and the z i s and the p i s are selected so that Aumption 2. is satisfied (i.e., the closed loop system is globally asymptotically stable). Remark 4.. Note that, if a Trackable Domain for the ramp were specified by r < M rad/sec with M > P α = 25 rad/sec, no linear controller, satisfying this specification, would exist. Although the closed loop system (4.), (4.2), and (4.5) does satisfy the steady state specifications, it is of interest to analyze its transient behavior as well. To accomplish this, consider the simplest controller of the form (4.5): C(s) =. (4.6) To check if Aumption 2. is satisfied, consider the following realizations of the plant (4.) and the controller (4.6) A p =, B p =, C = b/j /J [ ], (4.7) A c =, B c =, C c =, D c =. 8

10 Using these realizations and the method of [6], it is poible to show that Aumption 2. is satisfied with Q =. (4.8).3 The transients of (4.), (4.2), (4.6) shown in Figure 4., are obviously deficient: both the overshoot and the settling time are too large. A similar situation takes place for ramp inputs as well: transients are too long. To improve this situation, consider the controller (4.5) with C =, z =.5, and p = 5, i.e., and its realization C(s) = + s/.5 + s/5, (4.9) A c = 5, B c = 5(5/.5 ), C c =, D c = 5/.5. (4.) For this case, Aumption 2. is satisfied with.25.5 Q =.5. (4.) The transients of (4.), (4.2), (4.9), illustrated in Figure 4.2, show an improvement in overshoot but still long settling time. This, perhaps, is due to the fact that, as it follows from 5 rad (a) output y(t) (solid) and reference r(t) (dashed) sec 2 Nm (b) control input sat 2.5 (v(t)) sec Figure 4.: Transient response with controller C(s) = 9

11 5 rad (a) output y(t) (solid) and reference r(t) (dashed) sec 2 Nm (b) control input sat 2.5 (v(t)) sec Figure 4.2: Transient response with controller C(s) = +s/.5 +s/5 5 rad (a) output y(t) (solid) and reference r(t) (dashed) sec 2 Nm (b) control input sat 2.5 (v(t)) sec Figure 4.3: Transient response with controller C(s) = +s/.5 +s/5 Figure 4.2 (b), the control effort is underutilized. To correct this situation, consider the controller (4.5) with C =, z =.5, and p = 5, i.e., C(s) = + s/.5 + s/5. (4.2) With this controller, the transients are shown in Figure 4.3. As one can see, the settling time has improved considerably and is very close to its lower limit (about 2. sec), which is

12 imposed by the limitation of the control effort and the magnitude of the step. Also, Figure 4.3 (b) shows that the control effort is fully utilized and behaves almost like the time-optimal bang-bang controller. Similar behavior is observed for all magnitudes of steps up to rad and for ramps with slopes le than 2 rad/sec. We remark that, although the matrix Q for this controller had not been found, the global asymptotic stability of the system (4.), (4.2), (4.2) has been shown using the method of [8] and the aociated software posted at jmg/personal.html. 5. CONCLUSIONS This technical note investigates steady state tracking errors with respect to step, ramp and parabolic inputs in systems with saturating actuators. It shows that the roles of poles at the origin of the plant and the controller are different. Based on this, system types are introduced and the steady state errors and Trackable Domains are characterized in terms of these system types. As illustrated by an example, the results derived may be useful for selecting controllers and/or actuators that ensure desired trackable domains and steady state errors in feedback systems with saturating actuators. APPENDIX The proofs of Theorem 2. and 2.2 are based on the following lemmas. Lemma A.. Define and A = A p B c C p A c, B = B p, C = [ D c C p C c ], (A.) M = (A + BC) T (Q + 2 CT C) + (Q + 2 CT C)(A + BC), (A.2) where A p, B p, C p, A c, B c, C c, D c are defined in (2.2) and Q is defined in (2.3). Then, under Aumption 2., (a) A + BC is Hurwitz, (b) Q + 2 CT C >, (c) M, (d) (A + BC, M) is observable.

13 Proof: Using (A.), rewrite (2.2) with r = as ẋ = Ax + B sat α (v), v = Cx. (A.3) Under Aumption 2., global asymptotic stability of closed loop system (2.), (2.2), or equivalently the closed loop system (2.), (A.3), can be established using the Lyapunov function V (x) = x T Qx + If Cx α, this V (x) reduces to v sat α (τ)dτ. V (x) = x T (Q + 2 CT C)x, (A.4) (A.5) the derivative of V (x) along the trajectory of (2.), (A.3) becomes V (x) = x T Mx, (A.6) and (A.3) can be written as ẋ = (A + BC)x. (A.7) Since the closed loop system (2.), (A.3) is globally asymptotically stable, A + BC in (A.7) is Hurwitz, which proves (a). Since V (x) > and V (x), matrices Q + 2 CT C and 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 Aumption 2. hold. Then, for every asymmetric saturation, sat α, β (v) = sat α (v + β) β, with β < α, the closed loop system (2.2) with r =, i.e., ẋ p = A p x p + B p sat α, β (v), ẋ c = A c x c + B c ( y), y = C p x p, (A.8) v = C c x c + D c ( y), is globally asymptotically stable. Proof: Under Aumption 2., global asymptotic stability of closed loop system (2.), (2.2) can be established using the Lyapunov function V (x) = x T Qx + v sat α (τ)dτ, (A.9) 2

14 where x = [x T p x T c ] 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). ξ = Without lo of generality, aume < β < α. Introducing the state transformation α x and the notation µ = α v, rewrite (A.8) as α + β α + β ξ = Aξ + Bψ(µ), µ = Cξ. Here A, B, and C are defined in (A.) and ψ(µ) = γ, if γ < µ, ψ(µ) = µ, if α < µ < γ, α ( ) α + β α + β sat α, β α µ, i.e., (A.) (A.) α, if µ < α, α(α β) where γ = < α. α + β For the closed loop system (A.), (A.), select a Lyapunov function candidate as follows: V (ξ) = ξ T Qξ + µ ψ(τ)dτ, (A.2) where Q is the same as in (A.4). The derivative of V (ξ) along the trajectories of (A.), (A.) is V (ξ) = ξ T (A T Q + QA)ξ + ψ(cξ)(2b T Q + CA)ξ + CB[ψ(Cξ)] 2, (A.3) while the derivative of V (x) along the trajectories of (2.), (2.2) is V (x) = x T (A T Q + QA)x + sat α (Cx)(2B T Q + CA)x + CB[sat α (Cx)] 2. (A.4) Now we show by contradiction that V (ξ). Aume there exist ξ such that V (ξ ) >. This ξ must satisfy ψ(cξ ) sat α (Cξ ), i.e., Cξ > γ, otherwise, it would result in V (x) > at x = ξ. Define x = α γ ξ. Then, Cx > α, ψ(cξ ) = γ and sat α (Cx ) = α. Substituting x in (A.4) yields V (x ) = x T (A T Q + QA)x + sat α (Cx )(2B T Q + CA)x + CB[sat α (Cx )] 2 = (α/γ) 2 ξ T (A T Q + QA)ξ + α(2b T Q + CA)ξ α/γ + CBα 2 = (α/γ) 2 [ξ T (A T Q + QA)ξ + γ(2b T Q + CA)ξ + CBγ 2 ] (A.5) = (α/γ) 2 [ξ T (A T Q + QA)ξ + ψ(cξ )(2B T Q + CA)ξ + CB[ψ(Cξ )] 2 ] = (α/γ) 2 V (ξ ) >, 3

15 which contradicts Aumption 2.. Therefore, V (ξ) for all ξ. Next we show, again by contradiction, that the only solution of (A.), (A.) that is contained in {ξ V (ξ) = } is the trivial solution ξ(t). Let ξ(t), t, be a non-trivial solution of (A.), (A.) that satisfies V ( ξ(t)). Aume first C ξ(t) γ for all t. Since sat α (C ξ(t)) = ψ(c ξ(t)), system (2.), (2.2) and system (A.), (A.) are identical; therefore, x(t) = ξ(t) is a non-trivial solution of (2.), (2.2) as well. Moreover, as it follows from (A.3) and (A.4), V ( x(t)) = V ( ξ(t)). This contradicts Aumption 2.. Hence, ξ(t) cannot satisfy C ξ(t) γ for all t. Aume now that C ξ(t) γ for all t. Then, from (A.), (A.), ξ(t) = e At ξ() t + e Aτ Bγdτ. (A.6) Define x(t) = α γ ξ(t). Then, x(t) is a solution of (2.), (2.2), since C x(t) > α and x(t) = α γ ξ(t) = e At x() + t e Aτ Bαdτ. (A.7) Moreover, using the chain of equalities similar to (A.5), we obtain V ( x(t)) = (α/γ) 2 V ( ξ(t)). This again contradicts Aumption 2.. Hence, ξ(t) cannot satisfy C ξ(t) γ for all t either. Therefore, there must exist an interval (t, t 2 ), t t 2, such that C ξ(t) < γ for all t (t, t 2 ). In this interval, ξ(t) = e (A+BC)(t t ) ξ(t ), t (t, t 2 ), (A.8) and V ( ξ(t)) = ξ T (t)m ξ(t) =, t (t, t 2 ), (A.9) where M is defined in (A.2). Since M is negative semidefinite by Lemma A., it follows that M ξ(t) = Me (A+BC)(t t ) ξ(t ) =, t (t, t 2 ). (A.2) This, however, contradicts the observability of (A + BC, M), which must take place according to Lemma A.. Thus, ξ(t) is the only solution of (A.), (A.) that is contained in {ξ V (ξ) = }. Finally, it can be shown, again by contradiction, that V (ξ) > for all ξ and that V (ξ) as ξ. Therefore, the system (A.) and (A.), and, hence, (A.8) is globally asymptotically stable. 4

16 Proof of Theorem 2.: Three cases are poible: (a) C and P ; (b) C = and P ; (c) P =. (a) Aume C and P. Define x p, x c, y, and v as v = C + C P r, y = C P + C P r, x p = A p B p v, x c = A c B c (r y ). (A.2) Using ˆx p = x p x p, ˆx c = x c x c, ŷ = y y, ˆv = v v, (A.22) rewrite (2.2) as ˆx p = A pˆx p + B p sat α, v (ˆv), ˆx c = A cˆx c + B c ( ŷ), ŷ = C pˆx p, (A.23) ˆv = C cˆx c + D c ( ŷ). Note that condition (2.7) implies v < α. Therefore, Aumption 2. and Lemma A.2 ensure the global asymptotic stablility of (A.23). Hence, the steady state exists and y(t) converges to y, i.e., e step which proves (ii). exists. This proves (i). Moreover, by definition, e step e step = r y = r C P + C P r = is given by r + C P, (A.24) (b) C = implies that A c has an eigenvalue. Choose x c to satisfy and let A c x c =, C c x c = r P, v = r P, y = r, x p = A p B p v. Proceeding similarly to (a), it can be shown that if r < α P, then e step =. (c) P = implies A p has an eigenvalue. Choose x p to satisfy (A.25) (A.26) A p x p =, C p x p = r, (A.27) and let v =, y = r, x c =. Similarly to (a), it can be shown that e step completes the proof. =. This 5

17 r u s (t) s + e(t) v(t) y(t) - C(s) sat α (v) P(s) (a) r u s (t) + ė(t) v(t) ẏ(t) - sp(s) s C(s) sat α(v) (b) 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 ẏ 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, e ramp of system (a) can be analyzed using Theorem 2. applied to system (b). Indeed, Aumption 2. holds for system (b) since it is aumed to hold for system (a). Also, introducing P(s) = sp(s), Ĉ(s) = C(s), (A.28) s and noting that P = P, Ĉ =, (A.29) condition (2.7) for system (a) is equivalent to r < + P α for system (b). Thus, Ĉ Theorem 2. ensures the existence of steady state in system (b) and t ė(t) lim = r + P =. (A.3) Ĉ This implies that lim t e(t) = const., which proves (i). It follows from the proof of Theorem 2. that lim t v(t) < α. Therefore, e ramp be identical to that of the system with linear actuator, i.e., must e ramp = r lim t sp(s)c(s). (A.3) Since P = lim t sp(s), it follows that lim sp(s)c(s) = lim sp(s) lim C(s) = P C, t t t (A.32) 6

18 and This proves (ii). e ramp = r P C. The proof for the parabolic input case is similar. (A.33) REFERENCES [] D. S. Bernstein, and A. N. Michel, Chronological bibliography on saturating actuators, Int. J. Robust and Nonlinear Contr., vol. 5, no. 5, pp , 995. [2] A. Saberi, A. A. Stoorvogel, and P. Sannuti, Control of Linear Systems with Regulation and Input Constraints, London: Springer, 2. [3] T. Hu, Z. Lin, Control Systems with Actuator Saturation : Analysis and Design, Boston: Birkhäser, 2. [4] V. Kapila (editor), Actuator Saturation Control, New York: Marcel Dekker, 22. [5] K. S. Narendra and J. H. Taylor, Frequency Domain Criteria for Absolute Stability, New York: Academic Pre, 973. [6] F. Tyan and D. S. Bernstein, Global stabilization of systems containing a double integrator using a saturated linear controller, Int. J. Robust and Nonlinear Contr., vol. 9, pp , 999. [7] H. J. Sumann and Y. Yang, On the stabilizability of multiple integrators by means of bounded feedback controls, Proc. Conf. Dec. Contr., pp. 7 72, Brighton, U.K., 99. [8] J. Gonçalves, Quadratic surface Lyapunov functions in global stability analysis of saturation systems, Proc. Amer. Contr. Conf., Arlington, VA, June 2. [9] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, New Jersey: Prentice Hall,