China
Africa
Explore chapters and articles related to this topic
Feedback Equivalence of Nonlinear Control Systems: A Survey on Formal Approach
Published in Wilfrid Perruquetti, Jean-Pierre Barbot, Chaos in Automatic Control, 2018
We will say that the control-affine systems Σ and Σ˜ are locally feedback equivalent at x0 and x˜0, respectively, if ϕ is a local diffeomorphism satisfying ϕ(x0) = x˜0 and α and β are defined locally around x0. Note that local feedback equivalence is local in the state-space X but global in the control space U = ℝm.
Nonholonomic Behavior in Robotic Systems
Published in Richard M. Murray, Zexiang Li, S. Shankar Sastry, A Mathematical Introduction to Robotic Manipulation, 2017
Richard M. Murray, Zexiang Li, S. Shankar Sastry
Associated with a vector field, we define the flow of a vector field to represent the solution of the differential equation (7.7). Specifically, ϕtf(q) represents the state of the differential equation at time t starting from q at time 0. Thus ϕtf:ℝn→ℝn satisfies ddtϕtf(q)=f(ϕtf(q))q∈ℝn. A vector field is said to be complete if its flow is defined for all t. By the existence and uniqueness theorem of ordinary differential equations, for each fixed t, ϕtf is a local diffeomorphism of ℝn onto itself. Further, it satisfies the following group property: ϕtf∘ϕsf=ϕt+sf, for all t and s, where o stands for the composition of the two flows, namely ϕtf(ϕsf(q)).
Mathematical justification of a compressible bifluid system with different pressure laws: a continuous approach
Published in Applicable Analysis, 2022
Didier Bresch, Cosmin Burtea, Frédéric Lagoutière
Let us consider the flow generated by u: and observe that Hence, we get Thus, is a local -diffeomorphism for any . Next, with the help of (37), we write that for any we have Consequently, one has Consequently the application realizes a -diffeomorphism on for arbitrary We note that since u is 1-periodic, we also have In the next section, we make use of the following variant of the Reynolds transport theorem which we leave as an exercise to the reader.
Versal deformation of realisable Markov parameters
Published in International Journal of Control, 2019
Itziar Baragaña, Ferran Puerta
The map defined by is a local diffeomorphism at (ξ, I). That is, there exist open neighbourhoods of 0 in , of 0 in and of ξ in Σco such that the restriction of β, , is a diffeomorphism.
Output transformations and separation results for feedback linearisable delay systems
Published in International Journal of Control, 2018
F. Cacace, F. Conte, A. Germani
The idea is to use the predictor-based control proposed in Cacace et al. (2016b). To this end, we follow the approach described in Section 3.4. It is easy to verify that system (88)–(90) satisfies – and locally around it is diffeomorphic to the feedback linearisable system obtained by choosing as fictitious output the function In fact, the system has relative degree r1 = 1 with respect to h1 and r2 = 2 with respect to h2. Therefore is satisfied since r1 + r2 = 3. Moreover, is non-singular if x1 ≠ 0 and x2 ≠ 0, as required by , and the nonlinear map is a global diffeomophism, as required by . Under these conditions, system (88)–(90) is globally diffeomorphic to with The predictor-based controller of Theorem 2.4 requires R(y) to be a constant matrix. Since and are satisfied, we can apply Theorem 3.4 to obtain R(y) = I2 by means of the transformation (53), which defines a new fictitious output and the map which is a local diffeomorphism around . In the coordinates , system (88)–(90) assumes the form with from which