Sontag E.D. An algebraic approach to bounded controllability of linear systems. International Journal of Control, 39(1):181--188, 1984.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Proof: See [29, p. 164] 2 Theorem 3.5.8: A system generated by the equation (3. 9) is globally null controllable if and only if [A; B] is controllable, and A has all its eigenvalues lying in the closed left half plane (continuous time) or in the closed unit disk (discrete time) Proof: See [44] for the combined discrete and continuous time results. 2 Remark 3.5.9: It is important to note that polynomially unstable modes are in fact stabilisable using bounded controls; these are modes with eigenvalues on the stability boundary, and corresponding to non trivial Jordan blocks. Remark ....

....the continuous time case are ussually credited as first appearing in the text by E.B. Lee and L. Markus ( 29] although it was only proved in its full generality in the single input case. The papers of Brammer [5] and Schmitendorf and Barmish [42] removed the assumptions, and the paper by Sontag [44] gave an algebraic approach for both continuous time and discrete time systems. An important component of these works is the extension to more general constraint sets; such a level of generality is not of concern to the problems studied in this dissertation, however. In the paper by Brammer [5] ....

E.D. Sontag. An algebraic approach to bounded controllability of linear systems. Int. J. Control, 39(1):181--188, 1984.

....solution. One of the advantages of the proposed control scheme is that it can be considered analogous to a constrained linear GPC scheme if no model plant mismatch is considered. In this case, is possible to guarantee the overall closed loop stability if only input constraints are considered [34, 9]. More recently, two main results on stability for general constrained predictive control were given in [30] based on an infinite prediction horizon, and [20] by solving the problem with a state feedback law using linear matrix inequalities. However, if model plant mismatch is to be considered, ....

E. D. Sontag. An algebraic approach to bounded controllability of linear systems. International Journal of Control, pages 181--188, 1984.

....3. 2 Stabilization of linear discrete time systems with actuator constraints Consider the plant equation 13 and assume that the system is stabilizable, that all the eigenvalues of A are in the closed unit disk and that each component of u is magnitude bounded (ju i j ffl) For discrete systems, Sontag (1984) proved the existence of a feedback controller which globally asymptotically stabilizes this system. However, the construction of a stabilizing controller is difficult (Sontag and Yang 1991, Sussmann, Sontag and Yang 1992) For example, Teel (1992) showed that for a system with more than two ....

Sontag, E. (1984). An algebraic approach to bounded controllability of linear systems, International Journal of Control 39: 181--188.

....1. Proof: See, for example, 5] 2 We have shown, that with m properly chosen, Controller MPC globally asymptotically stabilizes any constrained stabilizable system with poles in the closed unit disk, using state feedback. When the inputs are constrained, i.e. u min u(k) u max 8 k, Sontag [4] showed that there does not exist a controller that globally stabilizes any system with poles outside the unit circle. 1 Thus, the MPC controller globally stabilizes all constrained systems for which a global stabilization is possible. Remark 3 Theorems 1, 2 and 3 hold as well if other norms ....

E. Sontag. An algebraic approach to bounded controllability of linear systems. International Journal of Control, 39:181--188, 1984.

....the recent textbook [8] or the overview paper [1] and the references therein. A quite general and unified description of the so called Anti Windup schemes is given for instance in [2] Analysis of constraint systems in terms of stability, controllability and feasibility is of interest as well [7, 9, 10, 11]. To solve the constraint control problem in a linear 1 Work supported by the german DFG (1996 1999 WAPprogram: topic Synthese optimaler Regler unter der Berucksichtigung von Beschrankungen und Robustheitsforderungen ) which is gratefully acknowledged. framework, one implicitly has to ....

E. D. Sontag. An algebraic approach to bounded controllability of linear systems. Int. J. of Control, 39(1):181-- 188, Jan. 1984.

....control problem involves basic issues suchascharacterization of the null controllable region and stabilizabilityonthenull controllable region. These issues have been focuses of study of and are nowwelladdressed for linear systems that are not exponentially unstable. For example, it is well known [2, 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 ....

.... 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 # #### ######### ## #### ## ### ## ### ## ##### ######## ##### ############ ####### ##### ##### ################# in [2, 8], is equivalenttothesystem 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 feedbacklaws was proposed in [11] for achain of integrators and was fully ....

[Article contains additional citation context not shown here]

E.D. Sontag, \An algebraic approach to bounded controllability of linear systems," #### ## #######, Vol. 39, pp. 181-188, 1984.

....same trajectory but arrive at the destination point at a later time. The automation of the on line trajectory replanning is an essential element of an automated Air Traffic Management System [1] The problem of stabilization in the presence of input constraints goes at least as far back as [2] In [3] it is shown that linear systems with poles in the right half plane can not be globally stabilized using bounded controls. In [2, 4] it is shown that for chains of integrators with more than two states, one must use nonlinear feedback in order to obtain global stabilization. For systems with ....

Sontag E.D. An algebraic approach to bounded controllability of linear systems. International Journal of Control, 39(1):181--188, 1984.

....to note that saturating inputs affect the classical controllability and stabilizability notions. For example, for the scalar system, x = x u juj M it is clear that regardless of the control law, any initial condition jx(0)j M can not be stabilized to the origin. This fact is documented in [6], where it is shown that linear systems with poles in the right half plane can not be globally stabilized using bounded controls. Therefore for systems with poles in the right half plane, stabilization results must be local. In this case one is interested in knowing what states are stabilizable. ....

Sontag E.D. An algebraic approach to bounded controllability of linear systems. International Journal of Control, 39(1):181--188, 1984.

....design is not so simple. However, under certain mild assumptions a design procedure may be formulated. In this regard, see [54] where sufficient conditions are given for the existence of piecewise linear controllers which regulate the nonlinear plant in a fixed number of time steps. Also, in [55] it is shown that bounded control inputs may be used to control the plant under certain conditions on linear or nonlinear plants in continuous or discrete time. In [59] stabilizability of a certain nonlinear time varying plant with output feedback is considered, and in [60] a similar problem is ....

E. D. Sontag. An algebraic approach to bounded controllability of nonlinear systems. International Journal of Control, 39:181--188, 1984.

....On the other hand, it was shown in [16] that a linear system subject to input saturation can be globally asymptotically stabilized by nonlinear feedback if and only if the system in the absence of saturation is asymptotically null controllable with bounded controls. This condition, as shown in [14,15], is equivalent to the system being stabilizable in the usual linear sense and having open loop poles in the closed left half plane. A nested feedback design technology for designing nonlinear globally asymptotically stabilizing feedback laws was proposed in [19] for a chain of integrators of ....

E.D. Sontag. An algebraic approach to bounded controllability of linear systems. Int. Journal of Control, 39:181--188, 1984.

....All the eigenvalues of A are located on the closed left half s plane; A2. The pair (A; B) is stabilizable; A3. The pair (A; C) is detectable. Remark 2.1. We should point out that assumptions A1 and A2 are equivalent to the notion of asymptotic null controllability that was introduced in [8] and [9]. Before stating the problem at hand, we have the following definitions. Definition 2.1. Semi global exponential stabilization via linear static state feedback) The system (2.1) 2.2) is semi globally exponentially stabilizable by linear static state feedback if for any a priori given bounded ....

E.D. Sontag, "An algebraic approach to bounded controllability of linear systems," International Journal of Control, vol. 39, pp. 181-188, 1984.

....and thrust limitations which must be taken into account while planning the reference trajectory. This is an essential element of the automated air traffic management system (ATMS) proposed by NASA [1] The problem of stabilization in the presence of input constraints goes as far back as [2] In [3], it is shown that linear systems with poles in the right half plane can not be globally stabilized using bounded controls. In [4] it is shown that for chains of integrators with more than two states, one must use nonlinear feedback in order to obtain global stabilization. For null controllable ....

Sontag E.D. An algebraic approach to bounded controllability of linear systems. International Journal of Control, 39(1):181--188, 1984.

....On the other hand, it was shown in [18] that a linear system subject to input saturation can be globally asymptotically stabilized by nonlinear feedback if and only if the system in the absence of saturation is asymptotically null controllable with bounded controls. This condition, as shown in [16,17], is equivalent to the system being stabilizable in the usual linear sense and having open loop poles in the closed left half plane. A nested feedback design technology for designing nonlinear globally asymptotically stabilizing feedback laws was proposed in [21] for a chain of integrators of ....

E.D. Sontag. An algebraic approach to bounded controllability of linear systems. Int. Journal of Control, 39:181--188, 1984.

....6. Proof. Theorem 6) First assume that there exists 2 E(A) IR . Then Lemma 7.3 proves that (12) is not satisfied and so Sigma ios is not observable. Next we prove the converse. If Re( 0 for all 2 oe(A) then there exists an admissible control function u such that x(t) 0 as t 1. See [6]. In this case, it is obvious that for any x 0 there is a control u and a time T such that jy u (T; x 0 )j 1 so Lemma 7.1 implies observability. Assume now that E(A) IR = and Re( 0 for 2 E(A) Then Lemma 7.4 implies that the conditions of implies that (12) holds. Hence Sigma ios is ....

Sontag, Eduardo D., "An algebraic approach to bounded controllability of linear systems," Int. J. Control, 1984, Vol. 39, No. 1, pp. 181-188.

....u .t ; x 0 :D y u .t ; x 0 # y 0 . In addition, we use the notation Q y.t :D y 0 .t ; x 0 # y 0 D ce t A x 0 . There are the following possibilities: 1: 3.A 6 0: Then there exists an admissible control function u 0 : 0; 1 U such that x u 0 .t ; x 0 0 . t 1 (See [5]) Because of condition 2, there exist T 1 0; 0 such that R T 1 0 U .ce t A B dt jy 0 j # 1 C 3 . Choose T 0 0 such that k ce T 1 A x u 0 .T 0 ; x 0 k . Next choose u 1 such that R T 1 0 ce t A Bu 1 .T 1 # t dt jy 0 j # 1 C 2 . This is possible ....

SONTAG E.D. "An algebraic approach to bounded controllability of linear systems", Int.J. Control, 39, pp. 181-188, 1984

....without loss of generality that we can assume that the local linearisation of the origin is Hurwitz. Typically, we are interested in the case where the linearisation is stable, but the open loop system is unstable, which ensures that the system is not globally stable (a long known result; see [8] for a recent treatment of the combined continuous and discrete time results) Assumption 4.1: It is assumed that A Gamma BK is Hurwitz. Additionally, we assume that the system has been normalised so that there exists a positive definite matrix P Gamma1 such that: A Gamma BK) 0 (A ....

E.D. Sontag. An algebraic approach to bounded controllability of linear systems. Int. J. Control, 39(1):181--188, 1984.

....u steering the origin to x 0 in finite time. PROOF: Since all eigenvalues of A have zero real part and the pair (A; B) is controllable, for each 0 there is some control v 0 for the system x = Ax Bu so that jv 0 (t)j for all t and which drives in finite time the origin to x 0 (see e.g. [12]) Since the range of oe contains a neighborhood of the origin, and using a measurable selection (Fillipov s Theorem) it is also true that there is a measurable control v which achieves the same transfer, for the system x = Ax Boe(u) Now let, along the corresponding trajectory, u(t) v(t) 0 ....

....has an eigenvalue on the imaginary axis. Let G p be the L p gain for (46) Fix any 0 so that G p 1. Then there is some m2n matrix F so that kF k and A B(F EF ) has an eigenvalue with positive real part, where E = diag(oe 0 1 (0) oe 0 m (0) This follows as in [12], or by noticing that one may first use a small F to make the system reachable from one input cf. 13] Remark 4.1.13 and then using Ackermann s formula for pole shifting. By the Small Gain Theorem, the system x = Ax Boe(f(x) F x u) x(0) 0 ; 53) obtained by feeding back F x, ....

Sontag, E.D., "An algebraic approach to bounded controllability of nonlinear systems," Int. J. Control 39(1984):181-188.

....asymptotically to the origin, that is, so that the solution of (1.1) converges to zero. It turns out, and in fact follows also from the results to be given, that if this property holds for some such U then it also holds for every U which contains the origin in its interior. Now, it is known (cf. [3]) that a system is ANCBC if and only if (1) the pair (A; B) is stabilizable or asycontrollable in the usual unconstrained sense (equivalently, the rank of [I Gamma A; B] is n for all complex with jj 1, cf. e.g. 5] exercise 4.4.7) and (2) the spectral radius of A is less or equal to one. ....

Sontag, E.D.,, An algebraic approach to bounded controllability of linear systems. Int. J. of Control, 39(1984): 181-188.

....of the associated linear system x = Ax Bu subject to constraints on control values; one needs that this system be asymptotically null controllable using arbitrarily small controls. The theory of controllability of linear systems with bounded controls is well studied; see for instance [3] [5], and references there, for the above characterization. Note that there may be nontrivial Jordan blocks in A corresponding to critical eigenvalues, so the system x = Ax may be unstable; this makes the problem more interesting. We only look at global problems; local stabilization can always be ....

Sontag, E.D., "An algebraic approach to bounded controllability of linear systems," Int. J. Control 39(1984): 181-188.

No context found.

Sontag E.D. An algebraic approach to bounded controllability of linear systems. International Journal of Control, 39(1):181--188, 1984.

No context found.

Sontag E.D. An algebraic approach to bounded controllability of linear systems. International Journal of Control, 39(1):181--188, 1984.

No context found.

Eduardo D. Sontag. An algebraic approach to bounded controllability of linear systems. 39:181--188, 1984.

No context found.

E. D. Sontag, "An algebraic approach to bounded controllability of linear systems," Int. J. Contr., vol. 39, no. 1, pp. 181--188, 1984.

No context found.

E.D. Sontag, An algebraic approach to bounded controllability of linear systems, International Journal of Control, vol. 39, pp. 181-188, 1984. 11

No context found.

E.D. Sontag, An algebraic approach to bounded controllability of linear systems, International Journal of Control, vol. 39, 1984, pp. 181-188.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC