Separable Subsystem Control

Chapter IV: Separable Subsystems

4.2 Separable Subsystem Control

Constructing the feedback linearizing controller in (3.6) requires the dynamics of the full-order system. However, in the case of large dimensional systems, the full dynamics may be unknown or may become computationally expensive, inhibiting feedback linearization.

Figure 4.1: Amputee-Prosthesis Separable System and Equivalent Subsystem.

(Left) Amputee-prosthesis separable system (blue), with separable prosthesis sub- system AMPRO3 (red). (Right) Equivalent prosthesis subsystem. (Middle) Control input from inverse dynamics of human-prosthesis motion capture walking data, determined with full-order system dynamics (blue) and with equivalent subsystem dynamics (red).

Control Law for Separable Subsystem

This section eliminates the need to know the full-order system dynamics for feedback linearization by constructing aseparable subsystem control law that only depends on subsystem dynamics. We begin by defining aseparable control system.

Definition 1: The affine control system(3.1)is aseparable control systemif it can be structured as

"

¤ 𝑥_𝑟

¤ 𝑥_𝑠

#

=

"

𝑓^𝑟(𝑥) 𝑓^𝑠(𝑥)

# +

"

𝑔^𝑟

1(𝑥) 𝑔^𝑟

2(𝑥) 0 𝑔^𝑠(𝑥)

# "

𝑢_𝑟 𝑢_𝑠

# , 𝑥_𝑟 ∈R^𝑛𝑟, 𝑥_𝑠 ∈R^𝑛𝑠, 𝑢_𝑟 ∈R^𝑚𝑟, 𝑢_𝑠 ∈R^𝑚𝑠,

(4.1)

where𝑛_𝑟 +𝑛_𝑠 =𝑛and𝑚_𝑟 +𝑚_𝑠 =𝑚.

Because of the structure of𝑔(𝑥) in (4.1), 𝑢_𝑟 only acts on part of the system. This motivates defining aseparable subsystemindependent of𝑢_𝑟.

Definition 2: For a separable control system (4.1), its separable subsystem is defined as

¤

𝑥_𝑠 = 𝑓^𝑠(𝑥) +𝑔^𝑠(𝑥)𝑢_𝑠, (4.2) which depends on the full-order system states𝑥 ∈R^𝑛.

Now, to construct a feedback linearizing control law for this separable subsystem, we construct output functions that solely depend on the subsystem states𝑥_𝑠 ∈ R^𝑛𝑠 and whose Lie derivatives solely depend on the subsystem (4.2).

Definition 3: For a separable subsystem (4.2) of the separable control system (4.1), a set of linearly independent output functions with vector relative degree

®

𝛾^𝑠 =(𝛾^𝑠

1, 𝛾^𝑠

2, . . . , 𝛾^𝑠

𝑚_𝑠)with respect to(4.1)areseparable subsystem outputsif they only depend on𝑥_𝑠 ∈R^𝑛𝑠,

𝑦^𝑠(𝑥_𝑠) ∈R^𝑚𝑠, (4.3)

and meet the following cross-term cancellation conditions for 𝑗 =1, . . . , 𝛾^𝑠

𝑖 −1and 𝑖=1, . . . , 𝑚_𝑠:

𝜕 𝐿

𝑗 𝑓^𝑠𝑦^𝑠(𝑥)

𝜕 𝑥_𝑟

𝑓^𝑟(𝑥) =0, (D3.1)

𝜕 𝐿

𝛾^𝑠

𝑖−1 𝑓^𝑠

𝑦^𝑠(𝑥)

𝜕 𝑥_𝑟 h

𝑔^𝑟

1(𝑥) 𝑔^𝑟

2(𝑥)i

= h 0 0

i

. (D3.2)

We use these outputs to introduce aseparable subsystem control lawin terms of the subsystem (4.2) alone.

Definition 4: For a separable subsystem (4.2) with separable subsystem outputs (4.3), we define a separable subsystem control law as the feedback linearizing control law

𝑢_ssc(𝑥) ≜ −(𝐿_𝑔^𝑠𝐿^®

𝛾^𝑠−1 𝑓^𝑠

𝑦^𝑠(𝑥)

| {z }

𝐴𝑠(𝑥)

)⁻¹(𝐿^®

𝛾^𝑠 𝑓^𝑠

𝑦^𝑠(𝑥)

| {z }

𝐿^∗

𝑓𝑠𝑦^𝑠(𝑥)

−𝜇_𝑠)

=−𝐴⁻¹

𝑠 (𝑥) (𝐿^∗

𝑓^𝑠𝑦^𝑠(𝑥) −𝜇_𝑠).

(4.4)

This control law is independent of the rest of the system dynamics 𝑓^𝑟(𝑥), 𝑔^𝑟

1(𝑥), and𝑔^𝑟

2(𝑥), but still depends on the full-order system states𝑥. We will address this dependence in subsequent results to develop an implementable form of this control law solely dependent on subsystem states and measurable inputs.

To compare this control law𝑢_ssc(𝑥)to𝑢_𝑠(𝑥), we constructseparable outputsfor the full-order system that include the separable subsystem outputs𝑦^𝑠(𝑥_𝑠)used for (4.4).

Definition 5: For a separable control system, a set of linearly independent output functions with vector relative degree𝛾®areseparable outputsif they are structured

as

𝑦(𝑥) =

"

𝑦^𝑟(𝑥) 𝑦^𝑠(𝑥_𝑠)

#

, 𝑦^𝑟(𝑥) ∈R^𝑚𝑟, 𝑦^𝑠(𝑥_𝑠) ∈R^𝑚𝑠, (4.5) and 𝑦^𝑠(𝑥_𝑠) are separable subsystem outputs with vector relative degree 𝛾®^𝑠. The remaining outputs 𝑦^𝑟(𝑥) have vector relative degree𝛾®^𝑟 and can depend on any of the system states𝑥. The number of subsystem outputs𝑚_𝑠 and the number of the rest of the outputs𝑚_𝑟 sums to𝑚, and𝛾®=( ®𝛾^𝑟,𝛾®^𝑠).

For the following theorem, we define the auxilary control input 𝜇as divided in the following form:

𝜇=

"

𝜇_𝑟 𝜇_𝑠

#

, 𝜇_𝑟 ∈R^𝑚𝑟, 𝜇_𝑠 ∈R^𝑚𝑠. (4.6) Theorem 1: For a separable control system (4.1), if the control law 𝑢(𝑥) = (𝑢_𝑟(𝑥)^𝑇, 𝑢_𝑠(𝑥)^𝑇)^𝑇 (3.6) is constructed with separable outputs (4.5) and auxiliary control input(4.6), then𝑢_𝑠(𝑥)=𝑢_ssc(𝑥).

Proof: We begin by relating the 3 components (𝐴⁻¹(𝑥), 𝐿^∗

𝑓

𝑦(𝑥), 𝜇)of𝑢(𝑥)to the components of𝑢_ssc(𝑥), (𝐴⁻¹

𝑠 (𝑥), 𝐿^∗

𝑓^𝑠

𝑦^𝑠(𝑥), 𝜇_𝑠). We are given 𝜇 = ★ 𝜇𝑠

by (4.6).

With condition (D3.2), we show:

𝐴(𝑥) = 













𝜕 𝐿^®

𝛾𝑟

−1 𝑓 𝑦^𝑟(𝑥)

𝜕 𝑥_𝑟

𝜕 𝐿^®

𝛾𝑟

−1 𝑓 𝑦^𝑟(𝑥)

𝜕 𝑥_𝑠

𝜕 𝐿^®

𝛾𝑠

−1 𝑓𝑠 𝑦^𝑠(𝑥)

𝜕 𝑥𝑟

𝜕 𝐿^®

𝛾𝑠

−1 𝑓𝑠 𝑦^𝑠(𝑥)

𝜕 𝑥𝑠















"

𝑔^𝑟

1(𝑥) 𝑔^𝑟

2(𝑥) 0 𝑔^𝑠(𝑥)

#

=













★ ★

0

𝜕 𝐿^®

𝛾𝑠

−1 𝑓𝑠 𝑦^𝑠(𝑥)

𝜕 𝑥_𝑠 𝑔^𝑠(𝑥)













=

"

★ ★

0 𝐴_𝑠(𝑥)

# ,

𝐴(𝑥)⁻¹=

"

★ ★

0 𝐴_𝑠(𝑥)⁻¹

# ,

where the final step is true because of the 0 corner block and properties of block matrix inversion. Similarly, we show:

𝐿^∗

𝑓𝑦(𝑥)=















𝜕 𝐿^®

𝛾𝑟

−1 𝑓

𝑦^𝑟(𝑥)

𝜕 𝑥𝑟

𝜕 𝐿^®

𝛾𝑟

−1 𝑓

𝑦^𝑟(𝑥)

𝜕 𝑥𝑠

𝜕 𝐿^®

𝛾𝑠

−1 𝑓𝑠 𝑦^𝑠(𝑥)

𝜕 𝑥_𝑟

𝜕 𝐿^®

𝛾𝑠

−1 𝑓𝑠 𝑦^𝑠(𝑥)

𝜕 𝑥_𝑠















"

𝑓^𝑟(𝑥) 𝑓^𝑠(𝑥)

#

=













★

𝜕 𝐿^®

𝛾𝑠

−1 𝑓𝑠 𝑦^𝑠(𝑥)

𝜕 𝑥𝑠

𝑓^𝑠(𝑥)













=

"

★ 𝐿^∗

𝑓^𝑠

𝑦^𝑠(𝑥)

# .

Putting these components together to construct𝑢(𝑥) yields 𝑢(𝑥)=−

"

★ ★

0 𝐴⁻¹

𝑠 (𝑥)

# "

★ 𝐿^∗

𝑓^𝑠

𝑦^𝑠(𝑥)

#

−

"

★ 𝜇_𝑠

#

=

"

★

−𝐴⁻¹

𝑠 (𝑥) (𝐿^∗

𝑓^𝑠𝑦^𝑠(𝑥) −𝜇_𝑠)

#

=

"

★ 𝑢_ssc(𝑥)

# , showing𝑢_𝑠(𝑥) =𝑢_ssc(𝑥) as defined in (4.4).

By Theorem 1, we can construct a stabilizing controller (4.4) for the subsystem independent of the rest of the system dynamics, identical to the portion of the controller constructed with the full-order system dynamics acting on the subsystem.

In the case of full-state feedback linearizable systems [39], this guarantees full- order system stability when (3.6) is applied to the rest of the system. This enables construction of stablemodel dependentcontrollers for separable subsystems without knowledge of the full-order system dynamics.

Equivalency of Subsystems

Although we now have a subsystem control law independent of the rest of the system dynamics, it still depends on the full-order system states𝑥 ∈ R^𝑛. Consider the case where the states 𝑥_𝑟 ∈ R^𝑛𝑟 cannot be measured. If we could construct an equivalent subsystem whose dynamics are a function of the subsystem states 𝑥_𝑠 ∈R^𝑛𝑠, and measurable inputsF ∈R^𝑛𝑓, we could calculate the subsystem control lawindependent of the full-order system states.

Consider another subsystem,

¤¯

𝑥_𝑠 = 𝑓¯^𝑠(X) +𝑔¯^𝑠(X)𝑢¯_𝑠 X =

"

¯ 𝑥_𝑠 F

#

=

"

𝑥_𝑠 F

#

∈R^𝑛¯,

(4.7)

where X is the state vector ¯𝑥_𝑠 = 𝑥_𝑠 augmented with an input F. Using the same separable subsystem outputsas (4.3), we define a control law ¯𝑢_𝑠(X)for the subsystem as:

¯

𝑢_𝑠(X) ≜ −(𝐿_𝑔¯𝑠𝐿^®

𝛾_𝑠−1

¯ 𝑓^𝑠

𝑦^𝑠(X)

| {z }

¯ 𝐴𝑠(X)

)⁻¹(𝐿^®

𝛾_𝑠

¯ 𝑓^𝑠

𝑦^𝑠( X)

| {z }

𝐿^∗

¯ 𝑓𝑠𝑦^𝑠(X)

−𝜇_𝑠)

=−𝐴¯⁻¹

𝑠 (X) (𝐿^∗

¯ 𝑓^𝑠

𝑦^𝑠(X) −𝜇_𝑠).

(4.8)

Theorem 2: For the subsystems (4.2) and(4.7), if∃ 𝑇 : R^𝑛 → R^𝑛¯ s.t. 𝑇(𝑥) = X and the following conditions hold,

𝑓^𝑠(𝑥) = 𝑓¯^𝑠(X), 𝑔^𝑠(𝑥) =𝑔¯^𝑠(X),

(T2) then 𝑢_𝑠(𝑥) = 𝑢¯_𝑠(X). Applying these to (4.2) and (4.7), respectively, results in dynamical systems such that given the same initial condition𝑥_𝑟0

𝑥_𝑠0

= h𝑥𝑟(𝑡₀)

𝑥𝑠(𝑡₀)

i yields solutions𝑥_𝑠(𝑡) =𝑥¯_𝑠(𝑡) ∀𝑡 ≥ 𝑡₀.

Proof. Since the subsystems have the same dynamics and outputs, the Lie derivatives comprising their control laws are also the same, hence

𝑢_𝑠(𝑥) =−𝐴⁻¹

𝑠 (𝑥) (𝐿^∗

𝑓^𝑠𝑦^𝑠(𝑥) −𝜇_𝑠)

=−𝐴¯⁻¹

𝑠 (X) (𝐿^∗

¯ 𝑓^𝑠

𝑦^𝑠(X) −𝜇_𝑠) =𝑢¯_𝑠(X).

With the same control law and dynamics, the closed-loop dynamics of the subsys- tems are the same:

¤

𝑥_𝑠 = 𝑓^𝑠(𝑥) +𝑔^𝑠(𝑥)𝑢_𝑠(𝑥)

= 𝑓¯^𝑠(X) +𝑔¯^𝑠(X)𝑢¯_𝑠(X)=𝑥¤¯_𝑠. Hence, given the same initial condition𝑥𝑟0

𝑥_𝑠0

= h

𝑥_𝑟(𝑡₀) 𝑥𝑠(𝑡₀)

i

, they have the same solution 𝑥_𝑠(𝑡)=𝑥¯_𝑠(𝑡) ∀𝑡 ≥ 𝑡₀.

By Theorems 1 and 2, we can construct a stabilizingmodel dependent controller, namely (4.8), for the subsystem (4.2) independent of the rest of the system dynamics and with measurable inputs.

Zero Dynamics. For full-state feedback linearizable systems, this control method will stabilize the full-order system. For partially feedback linearizable systems, we can apply this method to the feedback linearizable portion of the system. If the zero dynamics are stable, then we can guarantee full-order system stability, which will be proved through Lyapunov methods in the next chapter, Chapter 5.