Stabilization Using Sinusoids - Averaging and Control

Control of Underactuated 1-Homogeneous Systems with Drift

5.2 Averaging and Control

5.2.3 Stabilization Using Sinusoids

To summarize, we have obtained formulas for the response of 1-homogeneous control systems to an oscillatory control at some arbitrary order. We may analyze the effects of the control inputs on the expansions, leading to anα-parametrized form. Now, we must determine a stabilization feedback strategy. For conve- nience, order the Jacobi-Lie brackets as they appear in the averaged expansion (symmetric products are also brackets). LetYb_jdenote the Jacobi-Lie brackets, and letT^j(α)be their corresponding averaged coefficients.

With this ordering, the averaged equations can be put into the form:

z=X_S(z) + T^j(α)Yb_j(z)

=XS(z) +B(z)H(α), (5.45)

where the matricesB andHare

B(z) = [Yb1. . .YbN]andH(α) = [T¹. . .T^N]. (5.46) It is important to realize that Equation (5.45) will often represent a truncation of the full series expansion.

Theαparameters will be used to obtain control authority over the indirectly controllable states.

The dependence of the controller design on a local Lie algebra basis is problematic if the Jacobi-Lie brackets used to satisfy the small-time local (configuration) controllability vary pointwise. If the controller is to stabilize to different points, each with a unique basis spanning the Lie algebra, then multiple α-parametrizations are necessary [77]. For systems that are well-controllable, multiple parametizations can be avoided.

Stabilization via Discretized Feedback

We first consider feedback based on discrete sampling, while a continuous time version is developed below.

State error will be used as feedback to modulate theγ-parameters, converting the problem to periodic discrete feedback. The technique is very similar to the motion control algorithms, based on motion primitives, that use open-loop approximate inversion for open-loop stabilization and trajectory tracking [21, 24, 100].

However, averaging theory is explicitly used to prove stability. In the language of Floquet theory, our goal is to utilize feedback so as to create stable Floquet multipliers.

Theorem 37 Consider a system (5.16) which is small-time locally controllable atx^∗ ∈ E. Letu^k(t)be the set ofγ parametrized, T-periodic input functions wherek ={1, . . . , m}andγ ∈Rⁿ⁻^m. Letz(t), be the averaged system response to the inputs. Given the averaged system (5.45), assuming that themdirectly controlled states have been linearly stabilized and that the linearization of H from Equation (5.46), with respect toγ atγ = 0andz=x^∗, is invertible on the(n−m)dimensional subspace to control, there exists aK ∈R⁽ⁿ⁻^m)^×ⁿsuch that for

γ =−ΛKz(Tbt/Tc),

whereΛ⁽ⁿ⁻^m)^×⁽ⁿ⁻^m) is invertible andb·cdenotes the floor function, the average system response is stabi- lized.

proof

The proof was given in [160], but will be quickly sketched. Given the assumptions on the system, the averaged system (5.45) is controllable. Linearization with respect tozandγyields

z=Az+B ∂H

∂γ

γ=0

γ =Az+BΥγ.

Choosing γ constant over a period, the above system can be directly integrated to obtain a discrete, linear system

z(k+ 1) = ˆAz(k) + ˆBγ.

The assumptions imply thatBˆ has a pseudo-inverse,Λ, for the(n−m)-dimensional subspace to stabilize.

ChooseKso that the eigenvalues ofAˆ−Klie in the unit circle. This stabilizes the discrete system (i.e. the monodramy map), and the continuous system with piecewise constant feedback.

In the theorem it is assumed that the directly controlled state have been stabilized using continuous feedback, and a discretized feedback strategy is provided to stabilize the remaining states. At the end of each oscillation period, the error is computed and used to update the γ-parameters. The modified γ-parameters are held constant for the duration of the period, at which point a new error estimate is provided.

Stabilization via Continuous Feedback

The control design procedure outlined above required integration of the linearized dynamical model over a period of actuation, and a discretized feedback strategy. Corollary 3 implies that stability can also be determined from the Floquet exponents, i.e., the stability of the continuous autonomous averaged vector field. A version of Theorem 37 can be proven without discretizing the closed-loop system.

Theorem 38 Consider a system of the form (5.16) which is small-time locally controllable at x^∗ ∈ E.

Letu^a(t)be the corresponding set ofγ parametrized, T-periodic inputs functions wherea = 1. . . mand γ ∈Rⁿ⁻^m. Lastly denote byz(t), the averaged system response to the inputs. Given the averaged system (5.45), assuming that themdirectly controlled states have been linearly stabilized and that the linearization ofH, with respect toγ atγ = 0andz=x^∗, is invertible on the(n−m)dimensional subspace to control, then there exists aK∈R⁽ⁿ⁻^m)^×ⁿsuch that for

γ =−ΛKz(t)

whereΛ⁽ⁿ⁻^m)^×⁽ⁿ⁻^m)is invertible, we have stabilized the average system response.

proof

Same as Theorem. 37, without discretization.

Stabilization via Lyapunov Functions

In Section 2.3, the linearization of the autonomous Floquet vector field,Y, was emphasized as a means to prove stability of the original oscillatory flow. In reality, any method to demonstrate stability of the average can be applied; for example, using a Lyapunov function for the averaged system is permissible.

Definition 63 A stabilized truncated series expansion with respect to the Lyapunov function, V, for the vector field (2.32) is a truncated vector field that has the same stability property using the Lyapunov function, V, as any higher-order truncation of the vector field, and also the full series expansion of the vector field.

Recall, the averaged system response in Equation (5.45). Consider a parametrization forγ, e.g.,γ =p(z).

The system in Equation (5.45) may be expressed as

z=Z(z)≡X_S(z) +B(z)H◦p(z). (5.47) Theorem 39 Consider a system of the form (5.16) which is small-time locally controllable at z^∗ ∈ E.

Letu^a(t)be the corresponding set ofγ parametrized, T-periodic inputs functions wherea = 1. . . mand γ ∈Rⁿ⁻^m. Lastly, denote byz(t) the averaged system response to the inputs. Given the averaged system (5.45), assume that there exists an γ-parametrization and a Lyapunov function, V, such that (5.47) is a stabilized truncated series expansion with respect to the Lyapunov function, V. If the system in Equation (5.47) is shown to be asymptotically (exponentially) stable using the Lyapunov function, then the averaged system response is asymptotically (exponentially) stable.

proof

Follows the proof of 37, but without the discretization step. Instead of the linearization, use the Lyapunov function,V, to demonstrate stability.

Since the system (3.7) is nonlinear, the closed-loop system (5.47) will also contain nonlinear terms that complicate the Lyapunov analysis. In a local neighborhood of the equilibrium, some of the nonlinear terms do not affect stability and may be disregarded. Theorem 38 resolved the problem by linearizing the equations of motion; linearization emphasizes the dominant terms within a neighborhood of the equilibrium.

A general strategy for extracting the dominant stabilizing terms and removing the unnecessary nonlinear terms is needed for the Lyapunov technique to be useful.

Geometric Homogeneity. Geometric homogeneity may be used to identify higher-order terms that act as minor perturbations in a local neighborhood of the desired equilibriumz^∗. Recall that equilibria for systems with drift are elements of the zero section ofE.

Definition 64 Them^th-degree homogeneous truncation of a vector field, denotedTrunc^∆_m(·), is the trunc- tion obtained by removing all terms of homogeneous degree greater thanm.

In a local neighborhood of the equilibrium z^∗ ∈ E, the dominant terms will be those of homogeneous order less than or equal to0. Consequently, theγ-parametrization must result in state error feedback whose

homogeneous truncation of order0is stabilizing. The0-degree homogeneous truncation with respect to the dilation vector field,∆, results in the vector field,

z=Z₀ = Trunc^∆₀ (X_S(z) +B(z)H◦p(z)). (5.48) If there exists a Lyapunov function such that the system in Equation (5.48) is stable, then the averaged system is locally stable.

Theorem 40 Consider a system of the form (5.16) which is small-time locally controllable at z^∗ ∈ E.

Letu^a(t)be the corresponding set ofγ parametrized, T-periodic inputs functions wherea = 1. . . mand γ ∈Rⁿ⁻^m. Lastly denote byz(t), the averaged system response to the inputs. Given the averaged system (5.45), assuming that there exists anγ-parametrization and a Lyapunov function such that the homogeneous truncation in Equation (5.48) is asymptotically (exponentially) stable with respect to the Lyapunov function, then the actual system response is asymptotically (exponentially) stable.

proof

Same as Theorem 19, followed by invocation of Theorem 39.

Comments on the Stabilizing Controllers

Theorems 37-40 stabilize an equilibrium point of the averaged system. If theγ-parametrized control input functions do not vanish at the equilibriumq^∗, then by Theorem 9, the flow of the actual system stabilizes to an orbit (of sizeO()) around the fixed point. If, on the other hand, the input functions do vanish at the equilibrium, then Corollary 2 implies that the flow of the actual system stabilizes to the fixed point (i.e. the orbit collapses to the fixed point).

The states requiring continuous feedback use the averaged rather than the instantaneous values of the system for state feedback. Trajectories of the actual flow are related to the averaged flow by the Floquet mapping,

x(t) =P(t)(z(t)). (5.49)

We may solve for the averagez(t), using the current statex(t). SinceP(t)is given by a series expansion, we can easily compute its inverse.

For the discretized feedback strategy, the difference in the averaged and instantaneous indirectly controlled states is not a critical factor to consider due to the fact thatP(t)is periodic, i.e.,P(kT) = P(0) = Id, k∈Z⁺. The directly stabilized states do not use discretized feedback, and consequently do require the average to be used as feedback. If the Floquet mappingP(t) is ignored and instantaneous states x(t) are used in the feedback control of the directly controllable states, this averaging method will place an upper bound on the feedback gains. The oscillatory inputs should be faster than the natural stabilizing dynamics of the directly stabilized subsytem, otherwise there will be attenuation of the oscillatory signal (and consequently the feedback signal to the indirectly controlled states). With the exception of [118], this effect has not been discussed in prior presentations of feedback strategies that utilize averaging techniques [108, 136].

In practice, one may utilize averages computed in realtime as continuous feedback. The benefit of this latter approach is that the averaging process may serve to filter out noise in the sensor signals. It will also attenuate the feedback of external disturbances. As the continually computed average, x(t) =¯

1 T

t−Tx(τ) dτ, is not quite the same as z(t) = P⁻¹(t) (x(t)), there may be some differences. When performing averaging of sensed measurements, calculateP(t)to determine which states require averaging.

Recall that P(t) is a periodic function with a power series representation in. The states with dominant

oscillatory terms will require averaging, whereas the states that have no oscillatory terms, or ignorable, perturbative oscillatory terms, inP(t)will not require averaging.

Trajectory Tracking To track a trajectory, replace x(t) withx(t)−x_d(t); the system must be locally controllable along the trajectory. For the discretized feedback, the Nyquist criteria is a limiting factor in tracking a trajectory for the indirectly controlled states. As the trajectory evolves, the Jacobi-Lie brackets contributing to small-time local controllability may vary. The trajectory tracking feedback strategy is more complicated since multiple parametrizations wil be needed. If a system is well-controllable, then a single parametrization will suffice to track trajectories. Systems with group symmetris are well-controllable if and only if they are small-time locally controllable at the identity.

Configuration Control via Kinematic Motions. To recover the stabilization via kinematics motions from Bullo [21], only theα-parameters are used for feedback. The system must be initially at rest. Kinematic motions are those that return the sysstem to rest and the end of each actuation period.

Dalam dokumen Averaging and Control of Nonlinear Systems (Halaman 179-183)