## A general result on the stabilization of linear systems using bounded controls (1994)

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Venue: | IEEE Transactions on Automatic Control |

Citations: | 61 - 8 self |

### BibTeX

@ARTICLE{Sussmann94ageneral,

author = {Héctor J. Sussmann and Eduardo D. Sontag and Yudi Yang},

title = {A general result on the stabilization of linear systems using bounded controls},

journal = {IEEE Transactions on Automatic Control},

year = {1994},

volume = {39},

pages = {2411--2425}

}

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### Abstract

We present two constructions of controllers that globally stabilize linear systems subject to control saturation. We allow essentially arbitrary saturation functions. The only conditions imposed on the system are the obvious necessary ones, namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold. One of the constructions is in terms of a ”neural-network type ” one-hidden layer architecture, while the other one is in terms of cascades of linear maps and saturations.

### Citations

553 |
Applied Nonlinear Control
- Slotine, Li
- 1991
(Show Context)
Citation Context ...or rate limits. These systems cannot be naturally dealt with within the context of standard (algebraic) linear control theory, but are ubiquitous in control applications. To quote the recent textbook =-=[8]-=- (page 171): “saturation is probably the most commonly encountered nonlinearity in control engineering.” Mathematically, control questions become nontrivial, as only control values bounded by C are al... |

336 | Smooth stabilization implies coprime factorization
- Sontag
- 1989
(Show Context)
Citation Context ...jsN ffi for tsT , and our conclusion follows. 2 Remark 2.2 The terminology "SISS" should not be confused with the different --but closely related-- notion of "input to state stability&q=-=uot; (ISS) given in [10]-=- and other recent papers. It should also be possible to restate and prove the results given in this paper in terms of the ISS property, but the property called here SISS was exactly the one needed for... |

332 | Mathematical control theory, Deterministic Finite-Dimensional Systems
- Sontag
- 1990
(Show Context)
Citation Context ... m-input system 6 :sx = Ax +Bu. Assume without loss of generality that the first column b 1 of B is nonzero and consider the Kalman controllability decomposition of the system 6 1 :sx = Ax+b 1 u (see =-=[11], Lemma -=-3.3.3). We conclude that, after a change of coordinates y = T 01 x, 6 1 has the formsy 1 = A 1 y 1 + A 2 y 2 + �� b 1 u ;sy 2 = A 3 y 2 ; where (A 1 ; �� b 1 ) is a controllable pair. In these... |

70 |
Global stabilization and restricted tracking for multiple integrators with bounded controls
- Teel
- 1992
(Show Context)
Citation Context ...l laws that are implementable by feedforward nets with "hidden layers", rather than by the "perceptrons" represented by (1.2)). Recently, and motivated in part by [12] and [14], An=-=drew Teel showed in [15]-=- how, in the particular case of single-input multiple integrators, such combinations of saturations are indeed sufficient to obtain stabilizing feedback controllers. Here we obtain a general solution ... |

38 | An algebraic approach to bounded controllability of linear systems
- Sontag
- 1984
(Show Context)
Citation Context ...ts). (The theory of controllability of linear systems with bounded controls is a well-studied topic; see e.g. the fundamental paper [6], as well as the different, more algebraic approach discussed in =-=[9]-=-.) Notice that under Condition (ANCBC*) there may very well be nontrivial Jordan blocks corresponding to critical eigenvalues, so the systemsx = Ax need not be asymptotically stable or even Lyapunov-s... |

37 | Remarks on stabilization and input-to-state stability - Sontag - 1989 |

31 | Nonlinear output feedback design for linear systems with saturating controls
- Sontag, Sussmann
- 1990
(Show Context)
Citation Context ...ghly discontinuous. But by ignoring optimality questions, one may hope to find more regular and/or simpler controllers. Indeed, taking this point of view, we were able to obtain, in the previous work =-=[12]-=-, a general result on bounded stabilization by means of infinitely differentiable feedback laws. The result of [12] holds under the weakest possible conditions, namely, for ANCBC systems. Unfortunatel... |

29 |
A new design of constrained controllers for linear systems
- Gutman, Hagander
- 1985
(Show Context)
Citation Context ...tion is possible for systems that are neutrally stable (i.e. such that the Jordan form of A has no off-diagonal ones corresponding to imaginary eigenvalues), using the "Jurdjevic-Quinn" appr=-=oach (see [2]-=- and [7]). Thus it is natural to ask if simple control laws such as (1.2) can also be used for more general systems. This was negatively answered in a paper by A.T. Fuller as far back as the late 1960... |

25 |
In-the-large stability of relay and saturating control systems with linear controller
- Fuller
- 1969
(Show Context)
Citation Context ...it is natural to ask if simple control laws such as (1.2) can also be used for more general systems. This was negatively answered in a paper by A.T. Fuller as far back as the late 1960s. He showed in =-=[1]-=- that already for triple integrators such saturated linear feedback is not sufficient, at least under certain assumptions on the saturation oe. (A stronger negative result, which applies to basically ... |

25 | On finite gain stabilizability of linear systems subject to input saturation
- Chitour, Liu, et al.
- 1996
(Show Context)
Citation Context ...t lim t!1 e i (t) = 0. Let fl : [0; 1) ! IR n be a solution of (3.30) with v 1 = e 1 (t), v 2 = e 2 (t). Then lim t!1 fl(t) = 0. 2 Remark 3.7 The above proof of Lemma 3.2 is an adaptation of ideas of =-=[5]-=-. Using a similar technique, it can be shown that Corollary 3.5 also holds in higher dimensions, for controllable pairs (J; b) such that J is skew-symmetric, although in the general case it becomes mo... |

19 |
Null controllability of linear system with constrained controls
- Schmitendorf, Barmish
- 1980
(Show Context)
Citation Context ...e. all the uncontrollable modes of 6 have strictly negative real parts). (The theory of controllability of linear systems with bounded controls is a well-studied topic; see e.g. the fundamental paper =-=[6]-=-, as well as the different, more algebraic approach discussed in [9].) Notice that under Condition (ANCBC*) there may very well be nontrivial Jordan blocks corresponding to critical eigenvalues, so th... |

19 |
Using saturation to stabilize a class of single-input partially linear composite systems
- Teel
- 1992
(Show Context)
Citation Context ...bilizer for (8.1). As an example to illustrate the above procedure, we consider the systemsx 1 = x 2 ;sx 2 = x 3 ;sx 3 = tanh(x 4 ) ;sx 4 = u ; (8.6) which is of the form shown in Figure 4. (See also =-=[16], where th-=-e author considers the case when "tanh" in (8.6) is replaced by any function oe satisfying oe 0 (0) ? 0.) First, we need to find a smooth feedback that stabilizes the triple integratorsx 1 =... |

18 | On the stabilizability of multiple integrators by means of bounded feedback controls
- Sussmann, Yang
- 1991
(Show Context)
Citation Context ...feedback is not sufficient, at least under certain assumptions on the saturation oe. (A stronger negative result, which applies to basically arbitrary oe's, was more recently given, independently, in =-=[14]-=-.) The fact that linear feedback laws when saturated can lead to instability has motivated a large amount of research. (See for instance [3] and [4], and references therein, for estimates of the size ... |

15 | Global stabilization of linear systems with bounded feedback - Sontag, Yang - 1991 |

12 | Stabilization of linear systems with bounded controls
- Yang, Sussmann, et al.
- 1992
(Show Context)
Citation Context ...nstruction is shown to carry over to this case, and a separation principle is proved which allows the use of the saturated control design given earlier in the paper. Our result was first announced in =-=[17]-=-, where we considered a very special type of feedbacks for which the saturations are exactly linear near 0 and, when a system has a pure imaginary eigenvalue, a saturation with three different slopes ... |

8 |
Design of linear systems with saturating linear control and bounded states
- Kosut
- 1983
(Show Context)
Citation Context ...arbitrary oe's, was more recently given, independently, in [14].) The fact that linear feedback laws when saturated can lead to instability has motivated a large amount of research. (See for instance =-=[3]-=- and [4], and references therein, for estimates of the size of the regions of attraction that result when using linear saturated controllers.) Here we take a different approach. Rather than working wi... |

7 |
Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control
- Slemrod
- 1989
(Show Context)
Citation Context ...possible for systems that are neutrally stable (i.e. such that the Jordan form of A has no off-diagonal ones corresponding to imaginary eigenvalues), using the "Jurdjevic-Quinn" approach (se=-=e [2] and [7]-=-). Thus it is natural to ask if simple control laws such as (1.2) can also be used for more general systems. This was negatively answered in a paper by A.T. Fuller as far back as the late 1960s. He sh... |

3 |
Design of tracking systems subject to actuator saturation and integrator wind-up
- Krikelis, Barkas
- 1984
(Show Context)
Citation Context ...y oe's, was more recently given, independently, in [14].) The fact that linear feedback laws when saturated can lead to instability has motivated a large amount of research. (See for instance [3] and =-=[4]-=-, and references therein, for estimates of the size of the regions of attraction that result when using linear saturated controllers.) Here we take a different approach. Rather than working with linea... |

1 | Stabilization with saturated actuators, a worked example: F-8 longitudinal flight control
- Sontag, Yang
- 1993
(Show Context)
Citation Context ...f feedback controllers using very general types of activation (neuron characteristic) functions. For a detailed application of the results given in this paper to a model aircraft control example, see =-=[13]-=-. The paper is organized as follows. Our main result on state feedback is stated in Section 2 and proved in Section 4, using two technical lemmas proved in Section 3. In Section 5 we provide an algori... |