General averaging theory is merged with controllability analysis of underactuated nonlinear systems to derive exponentially stabilizing controllers. The stabilizing controllers derived using generalized averaging theory and STLC analysis can be used to stabilize both systems with and without drift.
Introduction
Review of Relevant Prior Work
- Series Expansions
- Averaging Theory
- Controllability of Mechanical and Nonlinear Systems
- Control of Underactuated Nonlinear Systems
A class of mechanical systems that generalizes much of the above work is the class of simple mechanical systems (SMS). Many of the solutions available for idle systems cannot be extended to the case of operated systems.
Contributions of This Thesis
One of the few studies of exponential stabilization of underactuated systems that has been extended from driftless to driven systems is [107]. Unactuated nonlinear systems with drift are more complicated, since satisfying LARC does not necessarily imply STLC [152].
Outline of Dissertation
Biomimetic control systems swim, walk, twist, and roll, demonstrating the universal nature of the control strategy found in this dissertation. Finally, Chapter 7 concludes with a summary of the results from the point of view of nonlinear and biomimetic and biomechanical control systems.
A Generalized Averaging Theory
Classical Averaging Theory
Ifxs(t) is a solution of (2.1) lying on the stable manifold of the periodic hyperbolic orbit γ, andzs(t). The proof of the stability part of this theorem is based on using the mean as a Poincar map of the current flow.
Evolution of Systems and the Chronological Calculus
- Series Expansions
- Elements of the Series Expansion
- Exponential Representation of a Flow and Perturbation Theory
Vt0,t(Xτ) is the Volterra series expansion of the flow corresponding to the time-varying vector field. The logarithm is at the heart of the series expansion solution for the flux approximation ΦXt0,t.
Averaging via Floquet Theory
- Linear Floquet Theory
- Nonlinear Floquet Theory
- Application to Averaging Theory
- Truncations of the Floquet Mapping
The proposition below relates the stability of the monodromy mapping to the stability of the original system. The role played by P0 is equivalent to that played by Θ in the monodromy map non-uniqueness debate.
A General Averaging Theory
- Higher-order Averaging
- A General Averaging Algorithm
Fourth-order averaging results start to get complicated due to the rate of growth of the averaging terms with each higher order. The difference between the roles of the mapping P(t) and the autonomous vector field was also made by Bogoliubov and Mitropolsky [17].
Perturbed Systems with Periodic Properties
- The Variation of Constants Transformation
- Perturbed Systems
- Highly Oscillatory Systems
The following two classes of perturbed systems to be studied as examples in Section 2.6 can be analyzed via the variation of constant transformation. The formula for variation of constants can be used to reproduce system (2.49) in the form required by averaging theory.
Examples
- Particle in an Electromagnetic Field
- Axial Crystal Lattice Channeling
- Observer Using Coupled Oscillators
After first-order averaging, the current of the vector field Y is approximated by the vector field. By observing the frequency of the (x2, p2) phase space oscillations, it should be possible to determine the energy of the first oscillator (x1, p1).
Averaging Theory and Dynamical Systems Theory
For diffeomorphisms that do not have a central manifold in the linearization, there are positive results related to the embedding problem [5, 88]. Therefore, the approach of this dissertation contrasts with the dominant approaches found in the body of literature dealing with the embedding problem through Floquet's theory.
Conclusion
Under weaker hypotheses, or in the appropriate sense, the fixed time maps of flows can generate all diffeomorphisms isotopically to the identity. This supports the idea that the logarithm can be used to obtain an autonomous vector field with the property that its fixed-time flow asymptotically approximates the diffeomorphism obtained from the fixed-time flow of a time-varying vector field sufficiently close flow to the identity, as per statement 2.
2.A Sanders and Verhulst Revisited
2.B Computation of Logarithm Series Expansions
For the fourth term and higher-order terms, the number of words begins to increase rapidly. Since Bernoulli numbers vanish for odd numbers greater than 1, those with depth 3 have vanishing coefficients.
2.C The Averaged Expansions
Control of Underactuated Driftless Nonlinear Systems
Control of Driftless Systems
- Averaging Theory for Control
- Sinusoidal Inputs for Indirect Actuation
- Stabilization Using Sinusoids
In equation (3.1), the functions ua(q, t) are control functions, which are decomposed into state feedback terms and time-periodic terms: ua(q, t) = fa(q) +va(t/). For a pair of oscillating inputs that create motion in the direction of the Jacobi-Lie bracket [Ya, Yb], the average coefficient in Lemma 2 is 2ωαab. If the inequality (3.25) is not satisfied, it will give rise to a Jacobi-Lie bracket of the form [Yi, Yj].
If the Lyapunov function can be used to demonstrate the stability of the homogeneous truncation (3.42), then at the local level the Lyapunov function stabilizes with respect to the average expansion of equation (3.38).
Mechanical Systems with Nonholonomic Constraints
- Systems with Nonholonomic Constraints
- Systems with Nonholonomic Constraints and Symmetries
By using the structure of the Ehresmann connection, the rank condition of the Lie algebra of equation (3.6) can be improved. The rank condition of the Lie algebra was simplified to the condition that the local curvature form and its covariant derivatives span the tangent to the fiber. For a Lie group, G, there exists a corresponding Lie algebra g defined using the tangent space of the group identity, TeG.
In the mean, the evolution of the basic space is trivial, since in the mean the basic variables are constant.
Examples
- Nonholonomic Integrator
- Hilare Robot
- Kinematic Car
This is the mathematical perspective of averaging found in the work of Sussmann and Liu [155, 93]. One consequence may be that a feasible controller for the canonical nonlinear controls could result in an infeasible controller for the actual system. For example, the nonlinear transformation to the canonical form results in a control law that significantly increases a control input, saturating the actual control. signal. An advantage of averaging theory over other methods is the existence of feedback control parameters.
Additional simulation snapshots can be found in Figure 3.11 for which it is possible to see the effects of the oscillatory actuation.
Conclusion
3.A Computing the Averaged Expansions
The final form of the third-order truncation of the averaged autonomous vector field Z is Trunc3(Y) =Λ(1)+2Λ(2)+3Λ(3).
Controllability and 1-Homogeneous Control Systems
Geometric Homogeneity and Vector Bundles
- Vector Bundles
- Geometric Homogeneity
- The Tangent Bundle and Vector Fields
It is known that the fiber of the tangent space to the vector space can be modeled by the vector space itself. Proposition 5 All smooth functions, Ψ : E1 → E2 of homogeneous order0, are equivalent to the mapping of the zero segment on the vector bundle E1 to E2. In the trivial division of vector fields into horizontal and vertical ones, the dilation vector field has the following properties.
For the systems we will study, the homogeneous order of the vector field Γ will not exceed 1, e.g.
Control of Dynamical Systems
- Configuration Controllability Revisited
- Conditions for Configuration Controllability
- Conditions for Small-Time Local Controllability
This is desirable since free symmetric algebra can be used to obtain properties of control vector fields, which will be useful for controllability calculations. The proof mirrors that of Lewis and Murray [85], however, the main steps of the proof will be outlined. From the primitive brackets, the only contributions to the Lie closure are those elements that lie in Br−1(X) and Br0(X), which correspond to the elements of the vertical and horizontal subspaces, respectively (using trivial decomposition).
The termXm+1 evaluates to the vector field Zlift which cannot occur on its own, it is part of the drift vector field.
Dissipation and Controllability
- Conditions for Configuration Controllability with Dissipation
Recall that to be an element of the equilibrium subspace, by Definition 44, the element must lie in the null section of E. An elementB∈Brl(X) contains brackets with more instances of the vector field X0 than the vector fields Xa,a= 1. The form of the Accessibility distribution on the Zero section for 1-homogeneous control systems with Dissipation.
Again, this is a generalization of the configuration controllability results found in Cort´es et al.
Examples
- The Tangent Bundle
- The Cotangent Bundle
- Constrained Mechanical Systems
- Constrained Mechanical Systems with Symmetry
This is behind much of the simplification found in the paper by Lewis and Murray [85]. It can be shown that the horizontal lifting of the constraint Ehresmann connection is still horizontal with respect to the trivial Ehresmann connection. It is normally assumed that the base space, M, through Q is fully controllable; (configurational) controllability of the fiber is the unknown.
The local form of the main connection, Aloc(r) :TrM →g, depends only on the base space, which means.
Conclusion
Controllability analysis for this case can be found in Ostrowski and Burdick, where part of the system evolves on the dual to the Lie algebra instead of the Lie algebra itself. In particular, the controllability analysis is reduced to the study of the curvature of the main connection as before, and also of the symmetric products. The calculations reduce not only to the tangent space of the vector bundle, T Q, but to a submanifold of this tangent space.
The role of dissipation has been investigated due to the desire to incorporate linearly dissipative forces into the equations of motion.
Control of Underactuated 1-Homogeneous Systems with Drift
Averaging of Oscillatory Systems with Drift
- Averaging and Geometric Homogeneity
The Floquet decomposition of equation (5.3) has the form necessary to obtain truncations of the infinite series expansions for Pe and Z. In particular, the proximity of the flow of truncated averages to the real flow (Theorems 11 and 13, the determination of stable orbits (Theorem 9), and the stabilization of fixed points (Corollary 2).The system can be transformed into the form required by the perturbation methods of averaging theory, according to the previous discussion.
After transforming the time and applying the variation of constants, we obtain the equivalent form of equation (5.3) for the 1-homogeneous vibration control systems of equation (5.16).
Averaging and Control
- Sinusoidal Inputs for Indirect Actuation
- Stabilization Using Sinusoids
The four permutations of input function choices result in the mean coefficient estimates found in Table 5.3. This section will consider only one of the average coefficients found in the average expansion; average coefficient, V(a,b,c)(2,2,1)(t), bracketed by Jacobi-Lie,. Failure to satisfy inequality (5.28) can excite a Jacobi-Lie bracket or symmetric product of the form,h.
Trajectories of the actual flow are related to the mean flow through the Floquet mapping.
Examples
- Nonholonomic Integrator
- Vibrational Control of a Passive Arm
Using the time-dependent oscillatory inputs forva(τ), the average form of the equations of motion is. The origin is a fixed point and the resulting control law will stabilize exactly at the origin, i.e., the orbit collapses to a point. The second-order equations of motion can be derived using the Lagrangian of the system, which is the kinetic energy minus the potential energy, L(q,q) =˙ 1.
In Figure 5.8, the same system is simulated without applying stabilizing feedback to the first joint.
Conclusion
Over time, the transient effect due to the initial conditions dies out, due to the stabilizing feedback to the first joint. Note that the transient response is attenuated due to the Rayleigh dissipation, but the angle of the first joint does not return to 0 degrees as in the case with feedback. Stability of the θ2 = π2 position applies to both cases, but we have seen that the domain of attraction is not the same.
Thus, classical averaging methods are valid only within a sufficiently small neighborhood of a fixed point.
5.A Computing the Averaged Expansions
Using the properties of the Jacobi-Lie bracket regarding geometric homogeneity, the bracket can be written as. Using the integration of products and assumption 2, the time multiplication coefficients can only be converted to integrals of the control inputs. Properties of the Jacobi-Lie bracket can be used to obtain simplifications of the third-order correction term.
