Constrained ILC for Human Motor Control

to track reference trajectory $\hat{\varvec{\varPhi }}_{\mathscr {P}}$ , where operator $G_{\mathscr {P}}$ is defined by (4.1). This objective assumes that a trajectory $\hat{\varvec{\varPhi }}_{\mathscr {P}}(t), \; t \in [0,T]$ is available. In the rehabilitation domain this is appropriate if the task is defined and explicitly presented to the patient as it was in Chap. 5. However, this is not possible when training more natural, everyday activities such as eating, washing or manipulating objects. To address this, the problem definition is now extended to encompass fully functional tasks.

6.1 Extended Task Representation

Fig. 6.1

Combined feedback and ILC control structure

To expand the task definition to capture the needs of human motor control, we define $0 = t_0 \le t_1 \le \cdots \le T_S = T$ distinct points in [0, T] which are deemed important to the task completion. These break the task down into S intervals in which one or more joints may be required to perform a synchronized movement. For example, time interval $[t_{j-1}, t_j]$ may correspond to the hand palm pushing a drawer along its runners, or the fingers holding a cup during a pouring movement. If $t_{j-1} = t_j$ is specified then the interval is an isolated time point, and may for example represent the time where the index finger makes contact with a light switch.

Given joint angle signal $\varvec{\varPhi }_{\mathscr {P}}$ defined over [0, T], we extract the single or linear combination of joints involved in any coordinated action by using the projection

$\begin{aligned} P&: \mathscr {L}_2^{n_p} [0,T] \rightarrow \mathscr {L}_2^{p_1} [0, t_1] \times \cdots \times \mathscr {L}_2^{p_{S}} [t_{S-1}, t_{S}] : \varvec{\varPhi }_{\mathscr {P}} \mapsto \left[ \begin{array}{c} (P \varvec{\varPhi }_{\mathscr {P}})_1 \\ \vdots \\ (P \varvec{\varPhi }_{\mathscr {P}})_S \end{array} \right] \end{aligned}$

(6.1)

with each component, $(P \varvec{\varPhi }_{\mathscr {P}})_j: \mathscr {L}_2^{n_p}[0,T] \rightarrow \mathscr {L}_2^{p_j}[t_{j-1}, t_j]$ , defined by

$\begin{aligned} ((P \varvec{\varPhi }_{\mathscr {P}})_j )(t) = P_j \varvec{\varPhi }_{\mathscr {P}}(t), \quad \quad t \in [t_{j-1}, t_j], \quad j = 1, 2, \ldots , S, \end{aligned}$

(6.2)

where

is a $p_j \times n_ p$ matrix of full row rank specifying the joint angles involved in the gesture or movement stipulated over time interval $[t_{j-1}, t_j]$ .

For ease of notation, the projected output $P \varvec{\varPhi }_{\mathscr {P}}$ is termed the “extended output” and denoted by $\varvec{\varPhi }_{\mathscr {P}}^e \in \mathscr {L}_2^{p_1} [0, t_1] \times \cdots \times \mathscr {L}_2^{p_{S}} [t_{S-1}, t_{S}]$ . We can also incorporate the projection into the system operator to yield the extended system operator $G_{\mathscr {P}}^e : \mathscr {L}_2^{p} [0,T] \rightarrow \mathscr {L}_2^{p_1} [0, t_1] \times \cdots \times \mathscr {L}_2^{p_{S}} [t_{S-1}, t_{S}]$ defined by

$\begin{aligned} \varvec{\varPhi }_{\mathscr {P}}^e = G_{\mathscr {P}}^e (\hat{\varvec{\varPhi }} + \varvec{v}) \; : \, G_{\mathscr {P}}^e (\hat{\varvec{\varPhi }} + \varvec{v}) = (P G_{\mathscr {P}}) (\hat{\varvec{\varPhi }} + \varvec{v}). \end{aligned}$

(6.3)

Using this extended task representation allows us to replace the ILC tracking problem (4.3) by the more general form

$\begin{aligned} \varvec{v}_{\infty } := \min _{\varvec{v}} J( \varvec{v} ), \;\;\;\;\; J( \varvec{v} ) = \big \Vert \hat{\varvec{\varPhi }}_{\mathscr {P}}^e - G_{\mathscr {P}}^e (\hat{\varvec{\varPhi }} + \varvec{v}) \big \Vert ^2, \end{aligned}$

(6.4)

where the extended reference trajectory and extended error are respectively

$\begin{aligned} \hat{\varvec{\varPhi }}_{\mathscr {P}}^e = \left[ \begin{array}{c} \hat{\varvec{\varPhi }}^{p_1}_{\mathscr {P}} \\ \vdots \\ \hat{\varvec{\varPhi }}^{p_S}_{\mathscr {P}} \end{array} \right] , \quad \varvec{e}_{\mathscr {P}}^e = \left[ \begin{array}{c} \hat{\varvec{\varPhi }}^{p_1}_{\mathscr {P}} - (P G_{\mathscr {P}})_1 (\hat{\varvec{\varPhi }} + \varvec{v}) \\ \vdots \\ \hat{\varvec{\varPhi }}^{p_S}_{\mathscr {P}} - (P G_{\mathscr {P}})_S (\hat{\varvec{\varPhi }} + \varvec{v}) \end{array} \right] = \hat{\varvec{\varPhi }}_{\mathscr {P}}^e - G_{\mathscr {P}}^e (\hat{\varvec{\varPhi }} + \varvec{v}). \end{aligned}$

(6.5)

Here $\hat{\varvec{\varPhi }}_{\mathscr {P}}^e$ contains the reference trajectories/points that must be followed at time points ${t}_{j-1} = t_j$ or over intervals $[{t}_{j-1}, {t}_j]$ . Note that the extended reference can be represented as the projection of a “virtual reference”, $\hat{\varvec{\varPhi }}_{\mathscr {P}}$ , denoted by $\hat{\varvec{\varPhi }}_{\mathscr {P}}^e = P \hat{\varvec{\varPhi }}_{\mathscr {P}}$ , however $\hat{\varvec{\varPhi }}_{\mathscr {P}}$ is not required in the control strategy. If the designer chooses

, and

then (6.4) collapses to the standard form (4.3).

6.2 Reduced Stimulation and Joint Subspaces

To reduce the number of degrees of freedom in problem (6.4) we can mimic the natural strategy of human motor control which involves a single neural command signal controlling multiple muscles. Each group of muscles working together is called a synergy, and the same muscle can potentially be employed within multiple synergies. To do this we introduce a set of $q \le m$ neural signals, denoted $\varvec{x} \in \mathscr {L}_2^{q}[0,T]$ . The mapping between $\varvec{x}$ and the m muscle stimulation signals $\varvec{u} \in \mathscr {L}_2^{m}[0,T]$ , can be represented at time t by $\varvec{u}(t) = \bar{X} \varvec{x}(t)$ . Here $\bar{X}$ is a $m \times q$ matrix with full column rank, with the jth column defining which muscles make up the jth synergy. The map between neural and muscle stimulation signals is therefore defined by $\varvec{u} = X \varvec{x}$ , with

$\begin{aligned} X : \mathscr {L}_2^{q}[0,T] \rightarrow \mathscr {X}[0,T] : \varvec{x} \mapsto \varvec{u}, \; \varvec{u}(t) = \bar{X} \varvec{x}(t). \end{aligned}$

(6.6)

Here $\mathscr {X}[0,T]$ is that subset of stimulation space $\mathscr {L}_2^{m}[0,T]$ which is achievable given the specified set of q synergistic muscle combinations, and is defined by

$\begin{aligned} \mathscr {X}[0,T] := \left\{ \varvec{u} = X \varvec{x}, \; \varvec{x} \in \mathscr {L}_2^{q}[0,T] \right\} \subset \mathscr {L}_2^{m}[0,T]. \end{aligned}$

(6.7)

The subspace $\mathscr {X}[0,T]$ is convex: to see this let $\tilde{\varvec{x}}$ , $\tilde{\varvec{y}} \in \mathscr {X}$ , then $\tilde{\varvec{x}}(t) = \bar{X} \varvec{x}(t)$ , $\tilde{\varvec{y}}(t) = \bar{X} \varvec{y}(t)$ then $\tilde{\varvec{x}}(t) + a (\tilde{\varvec{y}}(t) - \tilde{\varvec{x}}(t)) = X ( \varvec{x}(t) + a (\varvec{y}(t) - \varvec{x}(t)) ), \; a = [0,1]$ . Since $\varvec{x} + a (\varvec{y} - \varvec{x}) \in \mathscr {L}_2^{q}[0,T]$ it follows from (6.7) that $\tilde{\varvec{x}} + a (\tilde{\varvec{y}} - \tilde{\varvec{x}}) \in \mathscr {X}[0,T]$ .

Operator X restricts the stimulation signal $\varvec{u}$ to belong to a subspace $\mathscr {X}[0,T]$ of $\mathscr {L}_2^{m}[0,T]$ . However, we can also reduce the degree of freedom of control problem (6.4) by restricting the joint demand signal, $\varvec{v}$ , generated by ILC to belong to a suitable subspace of $\mathscr {L}_2^{p}[0,T]$ . To do this we introduce a set of q signals which represent synergies in joint space, and define the mapping between $\mathscr {L}^q_2[0,T]$ and the achievable subset of joint space by

$\begin{aligned} W : \mathscr {L}_2^{q}[0,T] \rightarrow \mathscr {W}[0,T] : \varvec{r} \mapsto \varvec{v}, \; \varvec{v}(t) = \bar{W} \varvec{r}(t). \end{aligned}$

(6.8)

where $p \times q$ matrix W has full column rank. The corresponding joint subspace is

$\begin{aligned} \mathscr {W}[0,T] := \left\{ \varvec{v} = W \varvec{r}, \; \varvec{r} \in \mathscr {L}_2^{q}[0,T] \right\} \subset \mathscr {L}_2^{p}[0,T]. \end{aligned}$

(6.9)

Embedding these subspaces into control problem (6.4) gives rise to:

Definition 6.1

Consider the system of Fig. 6.1 with control objective (6.4).

If the stimulation signal is restricted to subspace $\mathscr {X}[0,T]$ , ILC problem (6.4) is exchanged for

$\begin{aligned} \varvec{v}_{\infty } := \min _{\varvec{v}} J( \varvec{v} ), \;\;\;\;\; J( \varvec{v} ) = \big \Vert \overbrace{ \hat{\varvec{\varPhi }}_{\mathscr {P}}^e - G_{\mathscr {P}}^e (\hat{\varvec{\varPhi }} + \varvec{v}) }^{ \varvec{e}_{\mathscr {P}}^e} \big \Vert ^2, \quad \varvec{u}_k \in \mathscr {X}[0,T] \; \forall \; k \end{aligned}$

(6.10)

which simplifies to (6.4) if we impose the feedback controller structure

$\begin{aligned} K : \mathscr {L}^p[0,T] \mapsto \mathscr {X}[0,T] \; : K = X K_X, \; K_X : \mathscr {L}^p[0,T] \rightarrow \mathscr {L}^q[0,T]. \end{aligned}$

(6.11)

If instead the joint space demand signal is restricted to subspace $\mathscr {W}[0,T]$ , ILC problem (6.4) is exchanged for

$\begin{aligned} \varvec{v}_{\infty } := \min _{\varvec{v}} J( \varvec{v} ), \;\;\;\;\; J( \varvec{v} ) = \big \Vert \underbrace{ \hat{\varvec{\varPhi }}_{\mathscr {P}}^e - G_{\mathscr {P}}^e (\hat{\varvec{\varPhi }} + \varvec{v}) }_{\varvec{e}_{\mathscr {P}}^e} \big \Vert ^2, \quad \varvec{v}_k \in \mathscr {W}[0,T] \; \forall \; k. \end{aligned}$

(6.12)

The most general form of ILC problem we need to solve is therefore (6.12), which reduces to problem (6.4) if

, in which case $\mathscr {W} = \mathscr {L}^p_2[0,T]$ .

6.3 Extended ILC Framework

To address the problem (6.12) we extend Theorem 4.1 as follows:

Theorem 6.1

Consider the ILC update sequence

$\begin{aligned} \varvec{v}_{k+1}&= \varvec{v}_k + W L_k ( \varvec{e}_{k} )^e_{\mathscr {P}}, \quad \quad k = 0,1,\ldots , \quad \varvec{v}_0 \in \mathscr {W}[0,T]. \end{aligned}$

(6.13)

If learning operator $L_k : \mathscr {L}_2^{p_1} [0, t_1] \times \cdots \times \mathscr {L}_2^{p_{S}} [t_{S-1}, t_{S}] \rightarrow \mathscr {L}^{p}_2 [0,T]$ satisfies

$\begin{aligned} \big \Vert I - \bar{G}_{\mathscr {P}}^e |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} W L_k \big \Vert < 1 \quad \forall \; k \end{aligned}$

(6.14)

where the operator norm is induced from the inner product $\langle \cdot , \cdot \rangle$ , then

$\begin{aligned}&\lim _{k \rightarrow \infty } ( \varvec{e}_{k} )^e_{\mathscr {P}} = \varvec{0} \end{aligned}$

(6.15)

and, for $\varvec{v}_0$ chosen sufficiently close to $\varvec{v}_{\infty }$ , the ILC update converges to

$\begin{aligned}&\lim _{k \rightarrow \infty } (\hat{\varvec{\varPhi }} + \varvec{v}_k) = W L_{\infty } (\bar{G}_{\mathscr {P}}^e |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} W L_{\infty })^{-1} \hat{\varvec{\varPhi }}^e_{\mathscr {P}}, \quad \text {with} \quad \varvec{v}_k \in \mathscr {W}[0,T] \; \forall \; k. \end{aligned}$

(6.16)

Alternatively, if the learning operator $$L_k$$

satisfies

$\begin{aligned} \Vert I - L_{k} \bar{G}_{\mathscr {P}}^e |_{\varvec{v}_{\infty }} W \Vert < 1 \quad \quad \forall \; k \end{aligned}$

(6.17)

then, for $\varvec{v}_0$ chosen sufficiently close to $\varvec{v}_{\infty }$ , the ILC update converges to

$\begin{aligned}&\lim _{k \rightarrow \infty } (\hat{\varvec{\varPhi }} + \varvec{v}_k) = W ( L_{\infty } \bar{G}^e_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} W )^{-1} L_{\infty } \hat{\varvec{\varPhi }}^e_{\mathscr {P}} \quad \text {with} \quad \varvec{v}_k \in \mathscr {W}[0,T] \; \forall \; k. \end{aligned}$

(6.18)

Proof

Update (6.13) is equivalent to applying $\varvec{v}_k = W \varvec{r}_k$ to system (6.3) where

$\begin{aligned} \varvec{r}_{k+1} = \varvec{r}_k + L_k ( \varvec{e}_{k} )^e_{\mathscr {P}}, \quad \quad k = 0,1,\ldots , \quad \varvec{r}_0 \in \mathscr {L}_2^q[0,T] \end{aligned}$

(6.19)

and hence $\varvec{v}_k \in \mathscr {W}[0,T] \; \forall \; k$ . The error dynamics locally satisfy

$\begin{aligned} ( \varvec{e}_{k+1} )^e_{\mathscr {P}} = ( \varvec{e}_{k} )^e_{\mathscr {P}} - P \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{k}} W L_k ( \varvec{e}_{k} )^e_{\mathscr {P}} = (I - P \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{k}} W L_k) ( \varvec{e}_{k} )^e_{\mathscr {P}} \; \forall \; k \nonumber \end{aligned}$

so if (6.14) holds, the extended error converges monotonically to zero since

$\begin{aligned} \Vert ( \varvec{e}_{k+1} )^e_{\mathscr {P}} \Vert \le \Vert I - P \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{k}} W L_k \Vert \Vert ( \varvec{e}_{k} )^e_{\mathscr {P}} \Vert < \Vert ( \varvec{e}_{k} )^e_{\mathscr {P}} \Vert \; \forall \; k. \end{aligned}$

(6.20)

For $\varvec{v}_0$ sufficiently close to $\varvec{v}_{\infty }$ , then $\bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{k}} = \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }}$ and $L_k = L_{\infty } \; \forall k$ , and

$\begin{aligned} \varvec{v}_{k+1}&= \varvec{v}_0 + W L_{\infty } \sum ^k_{i = 0} ( \varvec{e}_{i} )^e_{\mathscr {P}} = \varvec{v}_0 + W L_{\infty } \sum ^k_{i = 0} \big ( I - P \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} W L_{\infty } \big )^i \nonumber \\[-0.4cm]&\quad \times \big ( \hat{\varvec{\varPhi }}^e_{\mathscr {P}} - P \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} ( \hat{\varvec{\varPhi }} + \varvec{v}_0 ) \big ) \nonumber \\&= W L_{\infty } \sum ^k_{i = 0} \big ( I - P \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} W L_{\infty } \big )^i \hat{\varvec{\varPhi }}^e_{\mathscr {P}} - \hat{\varvec{\varPhi }} \nonumber \\[-0.9cm] \nonumber \end{aligned}$

so that if (6.14) holds

$\begin{aligned} \lim _{k \rightarrow \infty } \varvec{v}_k = W L_{\infty } \big ( P \bar{G}_{\mathscr {P}} |_{ \hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} W L_{\infty } \big )^{-1} \hat{\varvec{\varPhi }}^e_{\mathscr {P}} - \hat{\varvec{\varPhi }}. \end{aligned}$

(6.21)

In addition, the input signals locally satisfy

$\begin{aligned} \varvec{r}_{k+1}&= \varvec{r}_{k} + L_{\infty } \big ( \hat{\varvec{\varPhi }}^e_{\mathscr {P}} - P \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} (\hat{\varvec{\varPhi }} + W \varvec{r}_k) \big ) \nonumber \\&= \big ( I - L_{\infty } P \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} W \big ) \varvec{r}_{k} + L_{\infty } \big ( \hat{\varvec{\varPhi }}^e_{\mathscr {P}} - P \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} \hat{\varvec{\varPhi }} \big ) \nonumber \\&= \big ( I - L_{\infty } P \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} W \big )^{k+1} \varvec{r}_0 \nonumber \\&\quad + \sum _{i = 0}^{k} \big ( I - L_{\infty } P \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} W \big )^i L_{\infty } \big ( \hat{\varvec{\varPhi }}^e_{\mathscr {P}} - P \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} \hat{\varvec{\varPhi }} \big ) \nonumber \\[-0.9cm] \nonumber \end{aligned}$

so that, if (6.17) is satisfied,

$\begin{aligned} \lim _{k \rightarrow \infty } \varvec{v}_k&= W \big ( L_{\infty } P \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} W \big )^{-1} L_{\infty } P \big ( \hat{\varvec{\varPhi }}_{\mathscr {P}} - \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} \hat{\varvec{\varPhi }} \big ) \nonumber \\[-0.2cm]&= W \big ( L_{\infty } P \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_{\infty }} W \big )^{-1} L_{\infty } \hat{\varvec{\varPhi }}^e_{\mathscr {P}} - \hat{\varvec{\varPhi }}. \end{aligned}$

(6.22)

$\square$

Within Theorem 6.1, the linearized extended plant operator is defined by:

Lemma 6.1

Around operating point $\tilde{\varvec{v}}$ the dynamics $\varvec{\varPhi }^e_{\mathscr {P}} = G^e_{\mathscr {P}} \varvec{v}$ are captured by the map $\bar{G}^e_{\mathscr {P}} |_{\tilde{\varvec{v}}} : \mathscr {L}_2^{p}[0,T] \rightarrow \mathscr {L}_2^{p_1} [0, t_1] \times \cdots \times \mathscr {L}_2^{p_{S}} [t_{S-1}, t_{S}]$ defined by

$\begin{aligned}&\bar{G}^e_{\mathscr {P}} |_{\tilde{\varvec{v}}} \varvec{v} = \left[ \begin{array}{c} (P \bar{G}_{\mathscr {P}} |_{\tilde{\varvec{v}}} )_1 \varvec{v} \\ \vdots \\ (P \bar{G}_{\mathscr {P}} |_{\tilde{\varvec{v}}} )_S \varvec{v} \end{array} \right] \end{aligned}$

(6.23)

where

$\begin{aligned} ((P \bar{G}_{\mathscr {P}} |_{\tilde{\varvec{v}}} )_j \varvec{v})(t) = {P_j} {\int _0^{t}} C(t) \varGamma ( t, \tau ) B( \tau ) \varvec{v} (\tau ) \mathrm {d} \tau , \quad t \in [{t_{j-1}}, {t_j}], \; j = 1, \ldots , S \end{aligned}$

(6.24)

in which A(t), B(t), C(t) and $\varGamma ( t, \tau )$ are defined in Lemma 4.2.

Theorem 6.2

Within (6.13), let the ILC operator be given by

$\begin{aligned} L_k = (\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*\big ( I + \bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W (\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*\big )^{-1}. \end{aligned}$

(6.25)

Setting $\varvec{v}_{k+1} - \varvec{v}_k = W (\varvec{r}_{k+1} - \varvec{r}_k) = W \varDelta \varvec{r}_k$ , this is equivalent to solving the underlying subspace problem

$\begin{aligned} \varDelta \varvec{r}_k := \min _{\varDelta \varvec{r}} \Big \{ \Vert \varDelta \varvec{r} \Vert ^2_{[R]} + \Vert (\varvec{e}_k)^e_{\mathscr {P}} - \bar{G}^e_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_k} W \varDelta \varvec{r} \Vert ^2_{Q} \Big \}, \quad \quad \varvec{r}_0 = \varvec{0}. \end{aligned}$

(6.26)

where $[R] = W^\top R W$ , with symmetric positive-definite weights Q and R.

If $\hat{\varvec{\varPhi }}^e_{\mathscr {P}} \in \text {im} ( G^e_{\mathscr {P}} W)$ then ILC operator (6.25) satisfies (6.14) and generates an input sequence satisfying

$\begin{aligned} \lim _{k \rightarrow \infty } \varvec{v}_k = \varvec{v}_{\infty }, \quad \varvec{v}_{\infty } := \min _{\varvec{v}} \left\| \varvec{v} \right\| ^2_R \;\; \text {s. t.} \;\; \varvec{e}^e_{\mathscr {P}} = \hat{\varvec{\varPhi }}^e_{\mathscr {P}} - G^e_{\mathscr {P}} (\hat{\varvec{\varPhi }} + \varvec{v}) = \varvec{0}. \end{aligned}$

(6.27)

If $\hat{\varvec{\varPhi }}^e_{\mathscr {P}} \notin \text {im} ( G^e_{\mathscr {P}} W)$ then ILC operator (6.25) satisfies (6.17) and generates an input sequence satisfying

$\begin{aligned} \lim _{k \rightarrow \infty } \varvec{v}_k = \varvec{v}_{\infty }, \quad \varvec{v}_{\infty } := \min _{\varvec{v}} \big \Vert \varvec{e}^e_{\mathscr {P}} \big \Vert ^2. \end{aligned}$

(6.28)

Proof

If $\hat{\varvec{\varPhi }}^e_{\mathscr {P}} \in \text {im} ( G^e_{\mathscr {P}} W)$ for all possible $\hat{\varvec{\varPhi }}^e_{\mathscr {P}}$ , it follows that $\text {ker}((\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*) = \{ \varvec{0} \} \; \forall \; k$ . If $\text {ker}((\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*) = \{ \varvec{0} \}$ the solution to (6.26) is $\varDelta \varvec{r}_k = L_k (\varvec{e}_k)^e_{\mathscr {P}}$ with $$L_k$$

given by (6.25) which satisfies (6.14) with limiting solution

$\begin{aligned} \varvec{v}_{\infty }&= W \varvec{r}_{\infty } = W (\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*\big ( \bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W (\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*\big )^{-1} \hat{\varvec{\varPhi }}^e_{\mathscr {P}} \end{aligned}$

(6.29)

corresponding to

$\begin{aligned} ( \varvec{e}_{\infty } )^e_{\mathscr {P}}&= \big ( I - \bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W (\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*\big ( \bar{G}^e | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W (\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*\big )^{-1} \big ) \hat{\varvec{\varPhi }}^e_{\mathscr {P}} = \varvec{0}. \nonumber \end{aligned}$

The Lagrangian associated with minimum energy problem (6.27) is just, with Lagrange multiplier $\varvec{\lambda } \in \mathscr {L}_2^{p_1} [0, t_1] \times \cdots \times \mathscr {L}_2^{p_{S}} [t_{S-1}, t_{S}]$ ,

$$$\begin{aligned} \mathscr {L}( \varvec{r}, \varvec{\lambda } )&= \Vert \varvec{v} \Vert _{R}^2 + 2 <\varvec{\lambda }, \hat{\varvec{\varPhi }}_{\mathscr {P}}^e - \bar{G}^e | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} \varvec{v}>\, = \Vert \varvec{r} \Vert _{[R]}^2 + 2 <\varvec{\lambda }, \hat{\varvec{\varPhi }}_{\mathscr {P}}^e - \bar{G}^e | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W \varvec{r}> \nonumber \end{aligned}$$” src=”/wp-content/uploads/2016/09/A352940_1_En_6_Chapter_Equ54.gif”></DIV></DIV></DIV>which has a stationary point when <SPAN id=IEq84 class=InlineEquation><IMG alt=$ and $\hat{\varvec{\varPhi }}_{\mathscr {P}}^e = \bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W \varvec{r}_{\infty }$ . The stationary point solution $\varvec{\lambda } = \big ( \bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W (\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*\big )^{-1} \hat{\varvec{\varPhi }}_{\mathscr {P}}^e$ is unique as is the defined input solution $\varvec{r}_{\infty } = (\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*\big ( \bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W (\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*\big )^{-1} \hat{\varvec{\varPhi }}_{\mathscr {P}}^e$ . This matches (6.29). If $\text {ker}((\bar{G}^e | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*) \ne \{ \varvec{0} \}$ the solution to (6.26) is $\varDelta \varvec{r}_k = L_k (\varvec{e}_k)^e_{\mathscr {P}}$ with

$\begin{aligned} L_k = \big ( I + (\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W \big )^{-1} (\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^* \end{aligned}$

(6.30)

which satisfies (6.17) with limiting solution, from (6.18),

$\begin{aligned} \varvec{v}_{\infty }&= W \varvec{r}_{\infty } = W \big ( (\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W \big )^{-1} (\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*\hat{\varvec{\varPhi }}^e_{\mathscr {P}} \end{aligned}$

(6.31)

with corresponding

$\begin{aligned} (\varvec{\varPhi }_{\infty })_{\mathscr {P}}^e&= \bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W \big ( (\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W \big )^{-1} (\bar{G}^e_{\mathscr {P}} | _{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*\hat{\varvec{\varPhi }}^e_{\mathscr {P}} \end{aligned}$

(6.32)

which is the minimizing solution of (6.28). $\square$

Lemma 6.2

Choosing

, $q/r \rightarrow \infty$ , the ILC update (6.25) realizes minimizing solutions (6.29) and (6.31) respectively in a single ILC iteration. In both cases the required term $W L_k ( \varvec{e}_k )^e_{\mathscr {P}}$ in (4.4) can be computed efficiently as the outcome, $\varDelta \varvec{v}^J$ , of J iterations of the computation

$\begin{aligned} \dot{\varvec{z}}(t)&= - A^\top (t) \varvec{z}(t) - C^\top (t) Q(t) P_i^\top P_i \big ( (\varvec{e}_k)_{\mathscr {P}}(t) - \bar{G}_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_k} \varDelta \varvec{v}^{j}_k (t) \big ), \nonumber \\ \varvec{z}(T)&= \varvec{0}, \quad t \in (t_{i-1}, t_i), \quad i = 1, \ldots , S \end{aligned}$

(6.33)

$\begin{aligned} \varDelta \varvec{v}_{k}^{j+1}(t)&= \varDelta \varvec{v}_{k}^{j}(t) + \alpha \bar{W} \bar{W}^\top R^{-1}(t) B^\top (t) \varvec{z}(t) \end{aligned}$

(6.34)

where J and $$$\alpha > 0$$” src=”/wp-content/uploads/2016/09/A352940_1_En_6_Chapter_IEq96.gif”></SPAN> are sufficiently large and small values respectively.</DIV></DIV><br /> <DIV id=FPar8 class=$

Proof

As $q/r \rightarrow \infty$ , update (6.25) and (6.30) respectively converge to

$\begin{aligned}&L_k = (\bar{G}^e_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^{*} \big ( \bar{G}^e_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_k} W (\bar{G}^e_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^{*} \big )^{-1}, \quad \text {and} \nonumber \\[-0.1cm]&L_k = \big ( (\bar{G}^e_{ \mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^{*} \bar{G}^e_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_k} W \big )^{-1} (\bar{G}^e_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^{*} \nonumber \end{aligned}$

which it is shown in [1] correspond to solutions of the minimum energy problem

$\begin{aligned}&\min _{\varDelta \varvec{r}_k} \big \Vert (\varvec{e}_k)^e_{\mathscr {P}} - P \bar{G}^e_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_k} W \varDelta \varvec{r}_k \big \Vert ^2, \quad \quad \varvec{r}_0 = \varvec{0} \end{aligned}$

(6.35)

using gradient based ILC [2]. This equates to $j = 1,2, \ldots , J$ iterations of update

$\begin{aligned}&\varDelta \varvec{r}_k^{j+1} = \varDelta \varvec{r}_k^j + \alpha ( P \bar{G}^e|_{\hat{\varvec{\varPhi }} + \varvec{v}_k} W )^*\big ( (\varvec{e}_k)^e_{\mathscr {P}} - P \bar{G}^e_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_k} W \varDelta \varvec{r}_k^j \big ) \nonumber \\[-0.1cm] \Rightarrow&\varDelta \varvec{v}_k^{j+1} = \varDelta \varvec{v}_k^j + \alpha W ( P \bar{G}^e_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_k} W )^*\big ( (\varvec{e}_k)^e_{\mathscr {P}} - P \bar{G}^e_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_k} \varDelta \varvec{v}_k^j \big ). \end{aligned}$

(6.36)

Operator $(P \bar{G}^e_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*= W^*(\bar{G}^e_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_k})^*P^*$ is defined by a relation of the form $\varvec{w} = (P \bar{G}^e_{\mathscr {P}} |_{\hat{\varvec{\varPhi }} + \varvec{v}_k} W)^*(\varvec{v}_1, \ldots , \varvec{v}_S)$ as the continuous solution of the costate equation

$\begin{aligned} \dot{\varvec{p}}(t)&= - A^\top (t) \varvec{p}(t) - C^\top (t) Q(t) P_j^\top \varvec{v}_j(t) \quad : \quad t \in [t_{j-1}, t_j), \quad \varvec{p}(T) = 0 \nonumber \\ \varvec{w}(t)&= \bar{W}^\top R^{-1}(t) B^\top (t) \varvec{p}(t). \end{aligned}$