Drift Vector Field Research Articles

A new Lyapunov design approach for nonlinear systems based on Zubov's method

The present research work proposes a new nonlinear controller synthesis approach that is based on the methodological principles of Lyapunov design. In particular, it relies on a short-horizon model-based prediction and optimization of the rate of “energy dissipation” of the system, as it is realized through the time derivative of an appropriately selected Lyapunov function. The latter is computed by solving Zubov's partial differential equation based on the system's drift vector field. A nonlinear state feedback control law with two adjustable parameters is derived as the solution of an optimization problem that is formulated on the basis of the aforementioned Lyapunov function and closed-loop performance characteristics. A set of system-theoretic properties of the proposed control law are examined as well. Finally, the proposed Lyapunov design method is evaluated in a chemical reactor example which exhibits nonminimum-phase behaviour.

On Local Structural Stability of Differential 1-Forms and Nonlinear Hypersurface Systems on a Manifold with Boundary

In this paper we consider smooth differential 1-forms and smooth nonlinear control-affine systems with (n−1)-inputs evolving on an n-dimensional manifold with boundary. These systems are called hypersurface systems under the additional assumption that the drift vector field and control vector fields span the tangent space to the manifold. We locally classify all structurally stable differential 1-forms on a manifold with boundary. We give complete local classification of structurally stable hypersurface systems on a manifold with boundary under static state feedback defined by diffeomorphisms, which preserve the manifold together with its boundary.

Underactuated ship tracking control: Theory and experiments

We consider complete state tracking feedback control of a ship having two controls, namely surge force and yaw moment. The ship model has similarities with chained form systems but cannot directly be transformed in chained form. In particular, the model has a drift vector field as opposed to the drift-free chained form systems. It is shown here that methods developed for tracking control of chained form systems still can be used for developing a tracking control law for the ship. Through a coordinate transformation the model is put in a triangular-like form which makes it possible to use integrator backstepping to develop a tracking control law. The control law steers both the position variables and the course angle of the ship, providing exponential stability of the reference trajectory. Experimental results are presented where the control law is implemented for tracking control of a model of an offshore supply vessel, scale 1:70. In the experiments the ship converges exponentially to a neighbourhood of the reference trajectory, and stays close with errors depending on factors as unmodelled dynamics, parameter uncertainty, measurement noise, thruster saturation, waves, currents and position measurement failures.

The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors

We consider stochastic differential equations in d-dimensional Euclidean space driven by an m-dimensional Wiener process, determined by the drift vector field f0 and the diffusion vector fields f1,...,fm, and investigate the existence of global random attractors for the associated flows φ. For this purpose φ is decomposed into a stationary diffeomorphism Φ given by the stochastic differential equation on the space of smooth flows on Rd driven by m independent stationary Ornstein Uhlenbeck processes z1,...,zm and the vector fields f1,...,fm, and a flow χ generated by the nonautonomous ordinary differential equation given by the vector field (∂Φt/∂x)−1[f0(Φt)+∑i=11fi(Φt)zti]. In this setting, attractors of χ are canonically related with attractors of φ. For χ, the problem of existence of attractors is then considered as a perturbation problem. Conditions on the vector fields are derived under which a Lyapunov function for the deterministic differential equation determined by the vector field f0 is still a Lyapunov function for χ, yielding an attractor this way. The criterion is finally tested in various prominent examples.

Minimax guaranteed cost control of uncertain non-linear systems

In this paper, a guaranteed cost control for non-linear uncertain systems is developed. The relationship between the idea of guaranteed cost control and dissipative feedback is first discussed which have been independently studied in the literature on robust control. It turns out that the two approaches are equivalent, and both presume the existence of a storage function (a Lyapunov function candidate) which guarantees the stability of the closed-loop system. The idea of guaranteed cost control is then applied to solve the robust control problem for non-linear systems with both drift vector field and input vector field uncertainties. Necessary and sufficient conditions are given for the solvability of this problem in terms of certain Hamilton-Jacobi Isaac's inequalities. The robustness properties of the controllers are also discussed. Finally, a passivity-based approach to the synthesis of guaranteed cost controllers is discussed.

On the regulation of a double effect evaporator: A trajectory planning and passivity approach

A dynamical feedback controller is proposed for the regulation of a simplified model of a double effect evaporator. The controller is based on feedback passivity considerations, complemented with a trajectory planning scheme in terms of the systems passive output. A drift vector field decomposition greately facilitates the direct application of the “energy shapping plus damping injection” feedback controller design methodology. The controller performance is enhanced with the aid of a trajectory planning scheme which effectively avoids singularity arcs in the state space, related to a loss of the relative degree of the passive output, while allowing for the satisfaction of physically meaningful state space and input space restrictions. The performance of the obtained controller design is evaluated by means of computer simulations

Feedback passivity of nonlinear multivariable systems

In this article, projection operators are used for obtaining a natural drift vector field decomposition that clearly exhibits the workless, the dissipative and the locally destabilizing forces acting within any nonlinear multivariable system equipped with a given positive definite storage function. A simple input coordinate transformation, which is shown to always exist, places the system into a Passivity Based Control Canonical form (PBCCF). Such a canonical form is ready for a direct application of the “energy shapping plus damping injection” controller design method.

On the curves that may be approached by trajectories of a smooth control affine system

In this paper, we give a characterization of the set of curves that may be approached by trajectories of a smooth control-affine nonlinear system, in the topology of uniform convergence. This characterization is in terms of the drift vector field and the distribution spanned by the Lie algebra generated by the control vector fields. The characterization is valid on open sets where this distribution has constant rank. These results also characterize the support of diffusion processes with smooth coefficients.

A general canonical form for feedback passivity of nonlinear systems

In this article projection operators are used for obtaining a natural canonical form, for non-linear single-input single-output systems, which clearly exhibits the conservative, the dissipative and the locally destabilizing forces present in the drift vector field of a given system, for which a storage function is known. When the system's natural 'energy management' structure is revealed, a suitable state-dependent input coordinate transformation makes the system passive or strictly output passive. A direct application of the 'energy shaping plus damping injection' controller design method, whch is a well known controller design option for passive systems, may then be attempted. A bioreactor controller design example is presented, for illustrative purposes, which includes computer simulations.

Circulation of lifted processes on trivial principal bundles

A Brownian motion {xt}t⩾0 on a compact Riemannian manifold M with a drift vector field X can be lifted to a diffusion process\(\left\{ {\tilde x_t } \right\}_{t \ge 0} \) on M × Tk corresponding to an ℝk valued smooth differential one-form A on M. The circulations (rotation numbers) of the lifted process\(\left\{ {\tilde x_t } \right\}_{t \ge 0} \) around the k circles of Tk are studied. By choosing a certain ℝk -valued differential one-form A, these circulations give the hidden circulation of {xt}t⩾0 in M and the rotation numbers of {xt}t⩾0 around some closed curves in M which generalize the first homology group H1(M,ℤ) of M.

Decompositions for control systems on manifolds with an affine connection

In this letter we present a decomposition for control systems whose drift vector field is the geodesic spray associated with an affine connection. With the geometric insight attained with this decomposition, we are able to easily prove some special results for this class of control systems. Examples illustrate the theory.

Resonance, stabilizing feedback controls, and regularity of viscosity solutions of Hamilton-Jacobi-Bellman equations

It is shown that no resonance and small time local controllability imply many standard necessary conditions for the existence of a continuous, asymptotically stabilizing (state) feedback control (ASFC) for ann-dimensional, real analytic, single input affine control system. Furthermore, no resonance means there exist local coordinates in which the drift vector field is linear, yielding a canonical form for the study of sufficient conditions and construction of an ASFC. The roles of resonance and the Kawski necessary condition are examined in detail, together with their implications on regularity of viscosity solutions of Hamilton-Jacobi-Bellman equations.

Universal stabilization of a class of nonlinear systems with homogeneous vector fields

Under relative-degree-one and minimum-phase assumptions, it is well known that the class L of finite-dimensional, linear, single-input ( u), single-output ( y) systems ( A,b,c) is universally stabilized by the feedback strategy u = Λ( λ) y, λ = y 2, where Λ is a function of Nussbaum type (the terminology “universal stabilization” being used in the sense of rendering /s(0/s) a global attractor for each member of the underlying class whilst assuring boundedness of the function λ(·)). A natural generalization of this result to a class H k of nonlinear control systems ( a,b,c), with positively homogeneous (of degree k ⩾ 1) drift vector field a, is described. Specifically, under the relative-degree-one ( cb ≠ 0) and minimum-phase hypotheses (the latter being interpreted as that of asymptotic stability of the equilibrium of the “zero dynamics”), it is shown that the strategy u = Λ( λ)/ vby/ vb k−1 y, dot λ = /vby/vb k+1 assures H k-universal stabilization. More generally, the strategy u = Λ( λ)exp(/ vby/ vb) y, dot λ = exp(/vby/vb)y 2 assures H -universal stabilization, where H = ∪ k ⩾ 1 H k.

Resonance and Feedback Stabilization

It is shown that no resonance and small time local controllability for an n-dimensional, real analytic, single input affine control system imply many standard necessary conditions for the existence of a continuous, asymptotically stabilizing (state) feedback control (ASFC). Furthermore, no resonance means there exist local coordinates in which the drift vector field is linear, yielding a canonical form for the study of sufficient conditions and construction of an ASFC. The roles of resonance and the Kawski necessary condition are examined in detail.

Stochastic Control Problems and Spherical Functions on Symmetric Spaces

A family of explicitly solvable stochastic control problems is formulated and solved in noncompact symmetric spaces. The symmetric spaces include all of the classical spaces and four of the exceptional spaces. The stochastic control problems are the control of Brownian motion in these symmetric spaces by a drift vector field. For each symmetric space a family of stochastic control problems is formulated by using spherical functions in the cost functionals. These spherical functions are explicitly described and are polynomials in suitable coordinates. A generalization to abstract root systems is given.

A liapunov-type theorem for nelson diffusions and nonattainability of nodes

Let us consider a Nelson Diffusion defined by the family of probability densities ρt and by the drift vector field b. Making use of the Liapunov function denotes the Stein regularization of : the distance function of , we prove the nonattainability of the nodal set of p under the conditions , where U is a nbh. of Zρ

Stochastic control problems and spherical functions on symmetric spaces

A family of explicitly solvable stochastic control problems is formulated and solved in noncompact symmetric spaces. The symmetric spaces include all of the classical spaces and four of the exceptional spaces. The stochastic control problems are the control of Brownian motion in these symmetric spaces by a drift vector field. For each symmetric space a family of stochastic control problems is formulated by using spherical functions in the cost functionals. These spherical functions are explicitly described and are polynomials in suitable coordinates. A generalization to abstract root systems is given.

Controllability of spacecraft systems in a central gravitational field

The configuration space for rigid spacecraft systems in a central gravitational field can be modeled by SO(3)/spl times/ IR/sup 3/, where the special orthogonal group SO(3) represents the attitude dynamics and IR/sup 3/ is for the orbital motion. The attitude dynamics of the spacecraft system is affected by the orbital elements through the well-known gravity-gradient torque. On the other hand, the effects of attitude-orbit coupling can also possibly be used to alter orbital motions by controlling the attitude. This controllability property is the primary issue of this paper. The control systems for spacecraft with either reaction wheels or gas jets being used as attitude controllers are proven in this study to be controllable. Rigorously establishing these results necessitates starting with the formal definitions of controllability and Poisson stability. It is then shown that if the drift vector field of the system is weakly positively Poisson stable and the Lie algebra rank condition is satisfied, controllability can be concluded. The Hamiltonian structure of the spacecraft model provides a natural route of verifying the property of weakly positive Poisson stability. Accordingly, the controllability is obtained after confirming the Lie algebra rank condition. Developing a methodology in deriving Lie brackets in the tangent space of T(SO(3)/spl times/IR/sup 3/), i.e., the second tangent bundle is thus deemed necessary. A general formula is offered for the computation of Lie brackets of second tangent vector fields in TT(SO(3)/sup m//spl times/IR/sup n/), in light of its importance in the fields of mechanics, robotics, optimal control, and nonlinear control, among others. With these tools, the controllability results can be proved. The analysis in this paper gives some insight into the attitude-orbit coupling effects and may potentially lead towards new techniques in designing controllers for large spacecraft systems. >

Stochastic processes in conformal Riemann-Cartan-Weyl gravitation

We construct the Riemann-Cartan geometries with torsion generated by the action of the conformal Weyl group. We study the wave operators associated to these structures, which, in addition to the usual Laplace-Beltrami operator, have a term which is a gradient vector field conjugate to the one-form given by the torsion potential derived from the Weyl group, and which we associate with a relativistic extension of the drift vector field in Nelson's construction of stochastic mechanics. In fact, our construction is valid for configuration spaces of any dimension. We sketch the construction of the stochastic processes on space-time generated by these operators, where the invariant measure is found to be defined by the conformal structure. We discuss briefly the relation with the theory of Dirichlet forms and D. Bohm's quantum potential in the theory of hidden variables, which in this setting acquire a gauge-geometric status previously unknown.

Geometric conditions for local controllability

Small time local controllability of a C 1 affine control system, with a generic drift vector field and n − 1 linearly independent input vector fields on an n-dimensional connected C 1 manifold M, is studied, showing that a necessary condition, namely the drift vector field belonging to the convex hull of the control vector fields at the given point, is also sufficient on an open dense subset H 1 of the set of points where it is verified. Local controllability at the points in the interior of H 1 is also considered when the input vector fields generate an involutive distribution, and a criterion based on the type of contract of H with the hyperplane spanned by the input vector fields is proved. An example shows a situation where the sufficient condition for small time local controllability of H. Sussmann [ SIAM J. Control Option. 25 (1987), 158–194] cannot be applied, but in which the method now presented distinguishes the possible cases.