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.< >
The problem of global stabilization is considered for a class of cascade systems. The first part of the cascade is a linear controllable system and the second part is a nonlinear system receiving the ...inputs from the states of the first part. With zero input, the equilibrium of the nonlinear part is globally asymptotically stable. In linear systems, a peaking phenomenon occurs when high-gain feedback is used to produce eigenvalues with very negative real parts. It is established that the destabilizing effects of peaking can be reduced if the nonlinearities have sufficiently slow growth. A detailed analysis of the peaking phenomenon is provided. The tradeoffs between linear peaking and nonlinear growth conditions are examined.< >
We prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point. Earlier results of Brunovsky, Hermes, Jurdjevic, Crouch and Byrnes, Sussmann and ...Grossmann, are shown to be particular cases of this result. Also, a number of new sufficient conditions are obtained. All these results follow from one simple general principle, namely, that local controllability follows whenever brackets with certain symmetries can be "neutralized," in a suitable way, by writing them as linear combinations of brackets of a lower degree. Both the class of symmetries and the definition of "degree" can be chosen to suit the problem.
We propose a definition of "regular synthesis" that is more general than those suggested by other authors such as Boltyanskii SIAM J. Control Optim, 4 (1966), pp. 326--361 and Brunovsky Math. ...Slovaca, 28 (1978), pp. 81--100, and an even more general notion of "regular presynthesis." We give a complete proof of the corresponding sufficiency theorem, a slightly weaker version of which had been stated in an earlier article, with only a rough outline of the proof. We illustrate the strength of our result by showing that the optimal synthesis for the famous Fuller problem satisfies our hypotheses. We also compare our concept of synthesis with the simpler notion of a "family of solutions of the closed-loop equation arising from an optimal feedback law," and show by means of examples why the latter is inadequate, and why the difficulty cannot be resolved byusing other concepts of solution---such as Filippov solutions, or the limits of sample-and-hold solutions recently proposed as feedback solutions by Clarke et al. IEEE Trans. Automat. Control, 42 (1997), pp. 1394--1407---for equations with a non-Lipschitz and possibly discontinuous right-hand side.
In a series of previous papers (cf. 20-23), we have developed a “primal” approach to the non-smooth Pontryagin Maximum Principle, based on generalized differentials, flows, and general variations. ...The method used is essentially the one of classical proofs of the Maximum Principle such as that of Pontryagin and his coauthors (cf. Pontryagin et al. 15, Berkovitz 1), based on the construction of packets of needle variations, but with a refinement of the “topological argument,” and with concepts of differential more general than the classical one, and usually set-valued.
In this article we apply this approach to optimal control problems with state space constraints, and at the same time we state the result in a more concrete form, dealing with a specific class of generalized derivatives (the “generalized differential quotients”), rather than in the abstract form used in some of the previous work.
The paper is organized as follows. In Sect. 2 we introduce some of our notations, and review some background material, especially the basic concepts about finitely additive vector-valued measures on an interval. In Sect. 3 we review the theory of “Cellina continuously approximable” (CCA) set-valued maps, and prove the CCA version – due to A. Cellina – of some classical fixed point theorems due to Leray-Schauder, Kakutani, Glicksberg and Fan. In Sect. 4 we define the notions of generalized differential quotient (GDQ), and approximate generalized differential quotient (AGDQ), and prove their basic properties, especially the chain rule, the directional open mapping theorem, and the transversal intersection property. In Sect. 5 we define the two types of variational generators that will occur in the maximum principle, and state and prove theorems asserting that various classical generalized derivatives – such as classical differentials, Clarke generalized Jacobians, subdifferentials in the sense of Michel–Penot, and (for functions defining state space constraints) the object often referred to as ∂ >xgin the literature – are special cases of our variational generators. In Sect. 6 we discuss the classes of discontinuous vector fields studied in detail in 24. In Sect. 7 we state the main theorem. The rather lengthy proof will be given in a subsequent paper.
A linear stabilizable, nonlinear asymptotically stable, cascade system is globally stabilizable by smooth dynamic state feedback if (a) the linear subsystem is right invertible and weakly minimum ...phase, and, (b) the only linear variables entering the nonlinear subsystem are the output and the zero dynamics corresponding to this output. Both of these conditions are coordinate-free and there is freedom of choice for the linear output variable. This result generalizes several earlier sufficient conditions for stabilizability. Moreover, the weak minimum-phase condition for the linear subsystem cannot be relaxed unless a growth restriction is imposed on the nonlinear subsystem.
We present a general necessary condition for separation of the reachable set of a Lipschitz control system from another given set of states, expressed in terms of an “approximating multicone” to the ...set in a sense that contains as special cases the Clarke and Mordukhovich cones. We then show how this separation result implies a strengthened form of the usual sufficient condition for local controllability along the reference curve and the necessary condition for optimality.
Telelearning in ophthalmology Sussmann, H.; Horsch, A.; Korda, W. ...
American journal of ophthalmology,
03/2005, Letnik:
139, Številka:
3
Journal Article
PDF
