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1 Null Controllability and Stabilization of Linear Systems Subject to Asymmetric Actuator Saturation Tingshu Hu, Achilleas N. Pitsillides & Zongli Lin Department of Electrical Engineering, University of Virginia Charlottesville, VA 93. th7f, Abstract This paper generalizes our recent results on the null controllable regions and the stabilizability of exponentially unstable linear systems subject to symmetric actuator saturation. The description of the null controllable region carries smoothly from the symmetric case to the asymmetric case. As to stabilization, we have to take a quite dierent approach since the development of our earlier relies mainly on the symmetric property of the vector field. Specifically, in this paper, we construct a Lyapunov function from a closed trajectory to show that this closed trajectory forms the boundary of the domain of attraction for a planar anti-stable system under the control of a saturated linear feedback. If the linear feedback is designed by the LQR method, then there is a unique limit cycle which forms the boundary of the domain of attraction. We further show that if the gain is increased along the direction of the LQR feedback, then the domain of attraction can be made arbitrarily close to the null controllable region. This design can be utilized to construct state feedback laws for higher order systems with two exponentially unstable poles. Introduction We consider the problem of controlling exponentially unstable linear systems subject to asymmetric actuator saturation. This control problem involves basic issues such as characterization of the null controllable region and stabilizability on the null controllable region. These issues have been focuses of study of and are now welladdressed for linear systems that are not exponentially unstable. For example, it is well-known [, 8] that such systems are globally null controllable with bounded controls as long as they are controllable in the usual linear system sense. In regard to stabilizability, it is shown in [9] that a linear system subject to actuator saturation can be globally asymptotically stabilized by smooth feedback if and only if the system is asymptotically null controllable with bounded controls (ANCBC, which, as shown Work supported in part by the US Oice of Naval Research Young Investigator Program under grant N in [, 8], is equivalent tothesystem being stabilizable in the usual linear sense and having open loop poles in the closed left-half plane. A nested feedback design technique for designing nonlinear globally asymptotically stabilizing feedback laws was proposed in [] for achain of integrators and was fully generalized in []. The notion of semi-global asymptotic stabilization on the null controllable region for linear systems subject to actuator saturation was introduced in [5]. The semi-global framework for stabilization requires feedback laws that yield a closed-loop system which has an asymptotically stable equilibrium whose domain of attraction includes an a priori given (arbitrarily large bounded subset of the null controllable region. In [5], it was shown that, for linear ANCBC systems subject to actuator saturation, one can achieve semi-global asymptotic stabilization on the null controllable region using linear feedback laws. On the other hand, the counterparts of the above mentioned results for exponentially unstable linear systems are less understood. Recently, we made an attempt to systematically study issues related to null controllable regions and the stabilizability on them of exponentially unstable linear systems subject to actuator saturation and gave a rather clear understanding of these issues [3]. Specifically, we gave a simple exact description of the null controllable region for a general anti-stable linear system in terms of a set of extremal trajectories of its time reversed system. We also constructed feedback laws that semi-globally asymptotically stabilize any linear time invariant system with two exponentially unstable poles on its null controllable region. This is in the sense that, for any a priori given set in the interior of the null controllable region, there exists a linear feedback law that yields a closed-loop system which has an asymptotically stable equilibrium whose domain of attraction includes the given set. One critical assumption made in [3] is that the actuator saturation is symmetric. The symmetry of the saturation function to a large degree simplifies the analysis of the closed-loop system, it, however, excludes the application of the results to many practical systems. The goal of this paper is to generalize the results of [3] to the case where the actuator saturation is asymmetric. We take a similar approach asin[3] tocharacterize

2 the null controllable region. In studying the problem of stabilization, we found the methods used in [3] to derive the main results not applicable to the asymmetric case, since the methods rely mainly on the symmetric property of the saturation function. In this paper, we propose a quite dierent approach to these problems for the asymmetric case. The proofs are sketched or omitted due to space limitation. Preliminaries and Notation Consider a linear system _x(t =Ax(t+bu(t; ( where x(t R n is the state and u(t R is the control. Given u < andu + >, let U m =fu u is measurable and u» u(t» u + ; 8t Rg A control signal u issaidtobeadmissible if u U m. In this paper, weareinterested in the control of the system ( by using admissible controls. Our first concern is the set of states that can be steered to the origin by an admissible control. Definition. A state x is said to be nullcontrollable if there exist a T [; and an admissible control u such that the state trajectory x(t of the system satisfies x( = x and x(t =. The set of all null controllable states is called the null controllable region of the system and is denoted by C. With the above definition, we have C = [ T [; ( Z T e Afi bu(fidfi u U m ( For simplicity, a linear system and the matrix A are said semi-stable if all the eigenvalues of A are in the closed left half plane; and anti-stable if all the eigenvalues of A are in the open right half plane. We recall a fundamental result from [, 6, 8] Proposition. Assume that (A; b is controllable. a If A is semi-stable, then C = R n. b If A is anti-stable, then C is a bounded convex open set containing the origin.» A c If A = with A R n n anti-stable A and A R n n semi-stable, and b is partitioned as» b b accordingly, then C = C R n where C is the null controllable region of the anti-stable system _x (t =A x + b u(t. Because of this proposition, we can concentrate on the study of null controllable regions of anti-stable systems. For this kind of systems, ρ Z C = e Afi bu(fidfi u U m ; (3 where C denotes the closure of C. We also to denote the boundary of a set. In this paper, we will derive a method for explicitly in Section 3. In the study of the null controllable regions we will assume, without loss of generality,that(a; b iscontrollable and A is anti-stable. Consider the time reversed system of ( _z(t =Az(t bv(t (4 Definition. A state z f is said to be reachable if there existt [; and an admissible control v such that the state trajectory z(t of the system (4 satisfies z(=and z(t =z f. The set of all reachable states is called the reachable region of the system (4 and is denoted byr. It is known that C of ( is the same as R of (4 (see, e.g., [6]. To avoid confusion, we will continue to use the notation x, u and C for the original system (, and z, v and R for the time-reversed system (4. 3 Null Controllable Regions In Section 3., we show that the boundary of the null controllable region of a general anti-stable linear system with saturating actuator is composed of a set of extremal trajectories of the time reversed system. The descriptions of this set are further simplified for systems with only real poles and for systems with complex poles in Sections 3. and 3.3, respectively. 3. Description of the null controllable regions We willcharacterize the null controllable region C of the system ( through studying the reachable region R of its time reversed system (4. Since A is anti-stable, we have R = = ρ ρ Z e Afi bv(fidfi Z e Afi bv(fidfi v U m v U m Noticing that e Afi = e A(fi,we see that a point z in R is a state of the time-reversed system (4 at t = by applying an admissible control v from to. Define the asymmetric sign function sgn a ( as sgn a (r = 8 < u + ; r>; (u + + u = ; r =; u ; r<

3 It can be verified that sgn a (r = u+ +u + u+ u sgn(r, where sgn( is the standard sign function. Theorem 3. ρ = e Afi b sgn a c e Afi b dfi c 6= (5 R is strictly convex. Moreover, for each z there exists a unique admissible control v Λ such that Z z Λ = e Afi bv Λ (fidfi (6 From Theorem 3., we see that if v is an admissible control and there is no c such thatv(t = sgn a (c e At b for t», then Z e Afi bv(fidfi and mustbeintheinterior of R. In what follows, we will simplify (5 and in terms of a set of trajectories of the time-reversed system (4. Denote E = Φ v(t = sgn a c e At b and for an admissible control v, denote Φ(t; v = ;t R c 6= Ψ ; (7 e A(tfi bv(fidfi (8 Since A is anti-stable, the integral in (8 exists for all t R, soφ(t; v iswell defined. If v(t =sgn a (c e At b, then Φ(t; v = = e A(tfi bv(fidfi Z e Afi b sgn a c e At e Afi b for any t R, i.e., Φ(t; v lies entirely An admissible control v such that Φ(t; v lies entirely is said to be extremal and such Φ(t; v anextremal trajectory. From Theorem 3., it can be shown that E is the set of extremal controls. Definition 3. v ;v Eare said to be equivalent, denoted by v ο v, if there exists an h R such that v (t =v (t h for all t R. The following theorem shows is covered by a minimal subset of the extremal trajectories. Theorem 3. Let E m ρebe such that for every v E, there exists a unique v E m such that v ο v. = fφ(t; v t R; v E m g (9 It turns out that for some classes of systems, E m can be easily described. For second order systems, E m contains only one or two elements, can be covered by no more than two trajectories; and for third order systems, E m corresponds to some real intervals. We will see later that for systems of dierent eigenvalue structures, the descriptions of E m can be quite dierent. 3. Systems with only real eigenvalues It follows from, for example, [6, p. 77], that if A has only real eigenvalues and c 6=,thenc e At b hasatmost n zeros. This implies that an extremal control can have at most n switches. It was shown in [3] that the converse is also true. Theorem 3.3 For the system (4, assume that A has only real eigenvalues, then a an extremal control has at most n switches; b any bang-bang control with n or less switches is an extremal control. It follows from Theorem 3.3 that E m can be chosen as the set of bang-bang controls with n or less switches and the first switch isatt =. Denote z e + = A bu + and z e = A bu, then we have, Observation is covered by two bunches of trajectories. The first bunch consists of trajectories of (4 when the initial state is z e + and the input is a bang-bang control that starts at t =with u and has n or less switches. The second bunch consists of the trajectories of (4 when the initial state is z e and the inputisabang-bang control that starts at t =with u + and has n or less switches. can be simply described in terms of the open-loop transition = (" n X i= sgn a (±( n I ±(u u + ( i e A(tti with t =. For second order = # A b t» t»t» ρe At z +e e A(tfi bu dfi t [; ] [ ρe At z e e A(tfi bu + dfi t [; ] 3.3 Systems with complex eigenvalues For a system with complex eigenvalues, the set E m is harder to determine. In what follows, we consider two important cases. Case. A R has apairofcomplex eigenvalues ± jfi, ; fi >. ;

4 The set of extremal controls is E = fv(t =sgn a (sin(fit + ;t R [; ßg It is easy to see that E m = fv(t = sgn a (sin(fit;t Rg contains only one element. Denote T p = ß fi, then e ATp = e Tp I. Let z s = e Tp u + e Tp u + A b; z + s = e Tp u + + e Tp u A b It can be verified that the extremal trajectory corresponding to v(t =sgn a (sin(fit is periodic with period T p, i.e., = e At z s e A(tfi bu + dfi t [;T p ] [ ρe At z +s e A(tfi bu dfi t [;T p ] Case. A R 3 3 has eigenvalues ± jfi and, with ; fi; >. a =. Then similar to Case, E m = fv(t =sgn a (k + sin(fit;t R k [; ]g b 6=. It can be shown that where E m = fu + ;u g[fv(t =sgn a (sin(fitg[e 3m E 3m = nv(t = sgn a ±e (t + sin(fit + o [; ß Plotted in Fig. are some extremal trajectories of the time-reversed system (4 with A = u + =,andu = ; B = ; 4 Domain of Attraction under Saturated Linear State Feedback Consider the open loop system _x(t =Ax(t+bu(t ( with admissible control u U m. A saturated linear state feedback is given by u = sat a (fx, where f Figure Extremal trajectories <. R n is the feedback gain and sat a ( is the asymmetric saturation function sat a (r = 8 <.5 u + ; r>u + ; r; r [u ;u + ]; u ; r<u Such a feedback is said to be stabilizing if A + bf is asymptotically stable. With a saturated linear state feedback applied, the closed loop system is _x(t =Ax(t+b sat a (fx(t ( Denote the state transition map of ( by i (t; x 7! x(t. The domain of attraction S of the equilibrium x = of ( is defined by n o S = x R n lim i(t; x = t! Obviously, S must lie within the null controllable region C of the system (. Therefore, a design problem is to choose a state feedback gain so that S is close to C. We refer to this problem as semi-global stabilization on the null controllable region. We will deal with anti-stable planar systems in this section. Consider the system (. Assume that A R is anti-stable. For the symmetric case where u = u +, it was shown in [3] is the unique limit cycle of the system (. This limit cycle is unstable for ( but is stable for the time-reversed system of (. So it can be easily obtained by simulating the time-reversed system. However, the method used in [3] to prove the uniqueness of the limit cycle relies on the symmetric property of the vector field. There is no obvious way to generalize the method to the asymmetric case. In this section, we will present a quite dierent approach to this problem. Actually, we will construct a Lyapunov function from the closed trajectory, and show that the Lyapunov function decreases in time as long as the trajectory starts from within the closed trajectory. Therefore, the open set enclosed by the closed trajectory is the domain of attraction..5

5 Lemma 4. The origin is the unique equilibrium point of the system ( and there is a closed-trajectory. Suppose that is a closed trajectory. By the index theory (e.g., see [4], must enclose the origin. Also, must have two intersections with one of the lines fx = u + or fx = u, or both of them. Otherwise there would be a closed trajectory completely in the linear region of the vector field. Proposition 4. Denote the region enclosed by the closed trajectory as Ω, then Ω is convex. The following theorem shows that under certain condition, a closed trajectory is the boundary of the domain of attraction. Theorem 4. Let be a closed-trajectory of the system (. Let the intersections of with the line fμa b μ Rg be x b and x b. If fx b ;fx b [u ;u + ], i.e., the two intersections x b and x b are between the two lines fx = u and fx = u +, =. Proof. Denote the region enclosed by as Ω. Since Ω contains the origin in its interior, we can define a Minkowski functional»(x =minl x flωg Clearly,»(x = for all x. Since is a trajectory and the vector field _x in ( is exists and is continuous on. Since Ω is bounded and convex, it follows 6= for all x. Note is the gradient of the function»(x, so it is perpendicular to the tangent of the curve =fx R»(x =g, which is _x. Therefore, (@»(x=@x _x =; 8x ( Define a Lyapunov function as V (x =» (x. It can be verified that for any constant >, and»(x =»(x; V (x = V x=x x=x (x=@x =»(x@»(x=@x, (x=@x exists and is continuous for all x R. With a detailed investigation of the vector field, it can be shown that for all x Ω, along the trajectory of the system (, _V (x =(@V (x=@x (Ax + b sat a (fx» It can also be shown that there exists no closed trajectory within Ω. Therefore, all the trajectories starting from within Ω will converge to the origin. It follows =. The condition fx b ;fx b [u ;u + ] in Theorem 4. is always true in a special case when the line fμa b μ Rg is in parallel to the line fx = u +. In the next section, we will show that if f is designed by the LQR method, then the line fμa b μ Rg is in parallel to the line fx = u +.In this case, any closed-trajectory is the boundary of the domain of attraction and hence there is a unique closed trajectory (a limit cycle. We will further show that the domain of attraction S can be made arbitrarily close to the null controllable region C by simply increasing the feedback gain. 5 Semi-Global Stabilization on the Null Controllable Region We will be focused on second-order anti-stable systems. The result can be easily extended to higher-order systems with two anti-stable mode with the technique in [3]. In this section, we continue to assume that A R is anti-stable and (A; b is controllable.to state the main result of this section, we need to introduce the Hausdor distance. Let X ; X be two bounded subsets of R n. Then their Hausdor distance is defined as where d(x ; X =max ~d(x ; X = sup n o ~d(x ; X ; d(x ~ ; X ; inf x X x X kx x k Here the vector norm used is arbitrary. Let P be the unique positive definite solution of the following Riccati equation, A P + PA Pbb P = (3 Then the origin is a stable equilibrium of the system _x(t =Ax(t+b sat a (kf x(t (4 for all k>5. Let S(k be the domain of attraction of the equilibrium x = of (4. Theorem 5. lim k! d(s(k; C =. Proof. For simplicity and without loss of generality, we assume that» A =» a ; a ;a > ; b = a Since A is anti-stable and (A; b iscontrollable, A; b can always be transformed into this form. With Λ this special form of A and b, we have f = a and A b» =. Hence, the line fμa b μ Rg is actually the line x =anditisbetween the two lines kf x = u + and kf x = u (x = u+ a and x k = u a k. Therefore, the condition in Theorem 4. is satisfied for all k > 5 and the closed-loop system has a unique limit cycle

6 which is the boundary of S(k. To visualize the for some k, are plotted in Fig., where the inner closed curve =, and the outer dashed one For convenience, we proceed the proof with the time reversed system of (4, _z(t =Az(t b sat a (kf z(t (5 Observe that is also the unique limit cycle of this system. Recall is formed by the trajectories of the system _z = Azbv one from z e + (orz s + toz e (orz s under the control v = u and the other from z e (orz s to z e + (orz s + underthecontrol v = u +. On the other hand, when k is suiciently large, the limit cycle must have two intersections with each of the lines kf z = u + and kf z = u. Suppose that the trajectory starts at the righthand side intersection with kf z = u, goes clockwise and intersects the two lines successively at time t ;t ;t 3, see the points z(;z(t ;z(t and z(t 3 in Fig.. We also note that from z( to z(t, v = sat a (kf z = u for the closed-loop system (5 and from z(t to z(t 3, v = u +. By comparing the two closed trajectories we can complete the proof by showing that as k!, z(;z(t 3! z e + (or z s +, and z(t ;z(t! z e (orz s ze (zs S(k=Γ z(t kf z=u + z(t 3 z(t Ω kf z=u Figure Illustration for the proof of Theorem 5. Example» 5. Consider» the open-loop system ( with 6 8 A =, b = 8 6, u 4 Λ = 5 and u + =. We have f = 66. In Fig. 3, the boundaries of the domains of attraction corresponding to dierent f = kf, k = 55; 6; 7; ;, are plotted from the inner to the outer. It is clear from the figure that the domain of attraction becomes larger as k is increased. The outermost dashed curve is@c. 6 Conclusions In this paper we have studied the problem of controlling a linear system subject to asymmetric actuator saturation. The null controllable region of such a C z( z e + (zs Figure 3 The domains of attraction under dierent feedback gains system is first characterized. Simple feedback laws are constructed to stabilize a system with no more than two exponentially unstable open-loop poles. The feedback law guarantees a domain of attraction that includes any given compact set inside the null controllable region. References [] J. Alvarez, R. Suarez and J. Alvarez, Planar linear systems with single saturated feedback," Systems & Control Letters, Vol., pp ,993. [] O. Hájek, Control Theory in the Plane, Springer- Verlag, 99. [3] T. Hu, Z. Lin and L. Qiu The controllability and stabilizability of exponentially unstable linear systems with saturating actuators," submitted for publication. [4] H. K. Khalil, Nonlinear Systsems, MacMillan, New York, 99. [5] Z. Lin and A. Saberi, Semi-global exponential stabilization of linear systems subject to `input saturation' via linear feedbacks," Systems and Control Letters, Vol., pp. 5-39, 993. [6] J. Macki and M. Strauss, Introduction to Optimal Control, Springer-Verlag, 98. [7] A. Saberi, Z. Lin and A.R. Teel, Control of linear systems with saturating actuators," IEEE Trans. Automat. Contr. Vol. 4, pp , 996. [8] E.D. Sontag, An algebraic approach to bounded controllability of linear systems," Int. J. Control, Vol. 39, pp. 8-88, 984. [9] E.D. Sontag and H.J. Sussmann, Nonlinear output feedback design for linear systems with saturating controls," Proc. 9th IEEE Conf. Decision and Control, pp , 99. [] H.J. Sussmann, E.D. Sontag, and Y. Yang, A general result on the stabilization of linear systems using bounded controls", IEEE Trans. Automat. Contr. Vol. 39, pp. 4-45, 994. [] A.R. Teel, Global stabilization and restricted tracking for multiple integrators with bounded controls", Syst. & Contr. Lett., Vol. 8, pp. 65-7, 99.