Adaptive least square control in discrete time of robotic arms

Abstract

In this paper, the trajectory tracking problem of robotic arms in discrete time is considered. To solve this problem, an adaptive least square controller is proposed. The uniform stability of the tracking error and parameters error for the aforementioned controller is guaranteed by means of a Lyapunov-like analysis. The effectiveness of the proposed controller is verified by on-line simulations.

Appendices

Appendix A: Proof of Lemma 1

From the first equation of (4) gives:

$$\begin{aligned} x_{1,k+2}=x_{1,k+1}+Tx_{2,k+1} \end{aligned}$$

(20)

Substituting the second equation of (4) into (20) gives:

$$\begin{aligned} x_{1,k+2}&= x_{1,k+1}+Tx_{2,k}+T^{2}M^{-1}(x_{1,k})\nonumber \\&\times \left[ u_{k}-G(x_{1,k})-C(x_{1,k},x_{2,k})x_{2,k}\right] \end{aligned}$$

(21)

Substituting the control (6) into (21) gives:

$$\begin{aligned} \widetilde{x}_{1,k+2}&= \widetilde{x}_{1,k+1}-K_{p}\widetilde{x}_{1,k+1} \nonumber \\ \widetilde{x}_{1,k+2}&= \left[ I-K_{p}\right] \widetilde{x}_{1,k+1} \end{aligned}$$

Obtaining the absolute value gives (7).

Appendix B: Proof of Lemma 2

Consider the case that $tr(\overline{V}_{k}^{M}-V_{1}^{M})\le r_{M}$, from (12) gives that $tr(V_{k}^{M}-V_{1}^{M})\le r_{M}$. Now consider $tr(\overline{V}_{k}^{M}-V_{1}^{M})>r_{M}$, from (12) gives:

$$\begin{aligned}&V_{k}^{M}=V_{1}^{M}+\frac{\overline{V}_{k}^{M}-V_{1}^{M}}{tr(\overline{V} _{k}^{M}-V_{1}^{M})}r_{M} \nonumber \\&tr(V_{k}^{M}-V_{1}^{M})=r_{M} \end{aligned}$$

then $V_{k}^{M}$ is always on the ball with center $V_{1}^{M}$ and radius $r_{M}$. The application of the above procedure in $V_{k}^{C}$ and $V_{k}^{G}$gives a similar result. Thus, (13) obtained using (12).

Appendix C: Proof of Lemma 3

From the first equation of (4) gives:

$$\begin{aligned} x_{1,k+2}=x_{1,k+1}+Tx_{2,k+1} \end{aligned}$$

Substituting into the second equation of (4) gives:

$$\begin{aligned} x_{1,k+2}&= x_{1,k+1}+Tx_{2,k}+T^{2}M^{-1}(x_{1,k})\nonumber \\&\times \left[ u_{k}-G(x_{1,k})-C(x_{1,k},x_{2,k})x_{2,k} \right] \end{aligned}$$

(22)

Substituting the control (8) into (22) gives:

$$\begin{aligned} M(x_{1,k})x_{1,k+2}&= M(x_{1,k})x_{1,k+1}-T\widetilde{M}(x_{1,k})x_{2,k} \nonumber \\&+T^{2}\widetilde{G}(x_{1,k})+T^{2}\widetilde{C}(x_{1,k},x_{2,k})x_{2,k}\nonumber \\&+\widehat{M}(x_{1,k})x_{1,d,k+2}-\widehat{M}(x_{1,k})x_{1,d,k+1}\nonumber \\&-\widehat{M}(x_{1,k})K_{p}\widetilde{x}_{1,k+1} \end{aligned}$$

(23)

where $\widetilde{M}(x_{1,k})=\widehat{M}(x_{1,k})-M(x_{1,k})$, $\widetilde{C}(x_{1,k},x_{2,k})=\widehat{C}(x_{1,k},x_{2,k})-C(x_{1,k},x_{2,k})$ and $\widetilde{G}(x_{1,k})=\widehat{G}(x_{1,k})-G(x_{1,k})$. It can be expressed as:

$$\begin{aligned}&M(x_{1,k})x_{1,k+2}-\widehat{M}(x_{1,k})x_{,1,d,k+2}\\&\quad =M(x_{1,k})x_{1,k+1}-\widehat{M}(x_{1,k})x_{1,d,k+1}-T\widetilde{M} (x_{1,k})x_{2,k}\\&\qquad +T^{2}\widetilde{G}(x_{1,k})\!+\!T^{2}\widetilde{C}(x_{1,k},x_{2,k})x_{2,k} \!-\!\widehat{M}(x_{1,k})K_{p}\widetilde{x}_{1,k+1} \end{aligned}$$

Adding and subtracting $\widehat{M}(x_{1,k})x_{1,k+2}$ and $\widehat{M}(x_{1,k})x_{1,k+1}$ gives:

$$\begin{aligned}&M(x_{1,k})x_{1,k+2}-\widehat{M}(x_{1,k})x_{1,d,k+2}\\&\qquad +\widehat{M}(x_{1,k})x_{1,k+2}-\widehat{M}(x_{1,k})x_{1,k+2} \\&\quad =M(x_{1,k})x_{1,k+1}-\widehat{M}(x_{1,k})x_{1,d,k+1}+\widehat{M} (x_{1,k})x_{1,k+1}\\&\qquad -\widehat{M}(x_{1,k})x_{1,k+1}-T\widetilde{M}(x_{1,k})x_{2,k}+T^{2} \widetilde{G}(x_{1,k})\\&\qquad +T^{2}\widetilde{C}(x_{1,k},x_{2,k})x_{2,k}-\widehat{M}(x_{1,k})K_{p} \widetilde{x}_{1,k+1} \end{aligned}$$

$$\begin{aligned} \widetilde{x}_{1,k+2}&= B\widetilde{x}_{1,k+1}+\left[ \widehat{M}^{-1} (x_{1,k})*\left( \widetilde{M}(x_{1,k})A_{k}\right. \right. \nonumber \\&\left. \left. \!\quad +\widetilde{C}(x_{1,k} ,x_{2,k})D_{k}+\widetilde{G}(x_{1,k})H_{k}\right) \right] \end{aligned}$$

(24)

where $B=I-K_{p}$, $A_{k}=x_{1,k+2}-x_{1,k+1}-Tx_{2,k}$, $D_{k}=T^{2}x_{2,k}$, $H_{k}=T^{2}1$, and $1=\left[ 1,\ldots ,1\right] ^{T}\in \mathfrak {R}^{n\times 1}$. Substituting (9), (10), and (23) into (24) gives:

$$\begin{aligned} \widetilde{x}_{1,k+2}&= B\widetilde{x}_{1,k+1}+\widehat{M}^{-1}(x_{1,k})\left( \widetilde{V}_{k}^{M}\varrho ^{M}\left( z_{M,k}\right) +\in _{k}^{M}\right) A_{k}\nonumber \\&\quad +\widehat{M}^{-1}(x_{1,k})\left( \widetilde{V}_{k}^{C}\varrho ^{C}\left( z_{C,k}\right) +\in _{k}^{C}\right) D_{k}\nonumber \\&\quad +\widehat{M}^{-1}(x_{1,k})\left( \widetilde{V}_{k}^{G}\varrho ^{G}\left( z_{G,k}\right) +\in _{k}^{G}\right) H_{k} \end{aligned}$$

(25)

where $\widetilde{V}_{k-1}^{M}=V_{k-1}^{M}-V_{1}^{M}$, $\widetilde{V} _{k-1}^{C}=V_{k-1}^{C}-V_{1}^{C}$ and $\widetilde{V}_{k-1}^{G}=V_{k-1} ^{G}-V_{1}^{G}$. (25) can be written as (14), it proves the lemma.

Appendix D: Proof of Theorem 1

Define the Lyapunov-like function as:

$$\begin{aligned} L_{k+1}\!=\!\frac{1}{2}tr\left\{ \widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}\right\} \!+\!\frac{1}{2}tr\left\{ \widetilde{V}_{k-1}^{T}P_{k-1} ^{-1}\widetilde{V}_{k-1}\right\} \quad \end{aligned}$$

(26)

where $tr\left\{ \widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}\right\}$ and$\ tr\left\{ \widetilde{V}_{k-1}^{T}P_{k-1}^{-1}\widetilde{V} _{k-1}\right\}$ are positive scalar numbers, the first difference is given by:

$$\begin{aligned} \Delta L_{k+1}&= \frac{1}{2}tr\left\{ \widetilde{x}_{1,k+2}\widetilde{x}_{1,k+2}^{T}\right\} -\frac{1}{2}tr\left\{ \widetilde{x}_{1,k+1} \widetilde{x}_{1,k+1}^{T}\right\} \nonumber \\&\quad +\frac{1}{2}tr\left\{ \widetilde{V}_{k}^{T}P_{k}^{-1}\widetilde{V} _{k}\right\} -\frac{1}{2}tr\left\{ \widetilde{V}_{k-1}^{T}P_{k-1} ^{-1}\widetilde{V}_{k-1}\right\} \nonumber \\&= \frac{1}{2}tr\left\{ \widetilde{x}_{1,k+2}\widetilde{x}_{1,k+2}^{T}\right\} -\frac{1}{2}tr\left\{ \widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}\right\} \nonumber \\&\quad -\frac{1}{2}tr\left\{ \left( \widetilde{V}_{k}-\widetilde{V}_{k-1}\right) ^{T}P_{k-1}^{-1}\left( \widetilde{V}_{k}-\widetilde{V}_{k-1}\right) \right\} \nonumber \\&\quad +\frac{1}{2}tr\left\{ \widetilde{V}_{k}^{T}\left( P_{k}^{-1}-P_{k-1} ^{-1}\right) \widetilde{V}_{k}\right\} \nonumber \\&\quad +tr\left\{ \widetilde{V}_{k}^{T}P_{k-1}^{-1}\left( \widetilde{V} _{k}-\widetilde{V}_{k-1}\right) \right\} \end{aligned}$$

(27)

Substituting (14) and using (15) and (16) into $\widetilde{x}_{1,k+2}\widetilde{x}_{1,k+2}^{T}-\widetilde{x}_{1,k+1} \widetilde{x}_{1,k+1}^{T}$ gives:

$$\begin{aligned}&\widetilde{x}_{1,k+2}\widetilde{x}_{1,k+2}^{T}-\widetilde{x}_{1,k+1} \widetilde{x}_{1,k+1}^{T}\nonumber \\&\quad =-\widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}\nonumber \\&\qquad +\left( \left[ B\widetilde{x}_{1,k+1}+\widehat{M}^{-1}(x_{1,k})\left( \widetilde{V}_{k}^{T}\varrho +\in _{k}\right) \beta _{k}\right] \right. \nonumber \\&\qquad *\left. \left[ B\widetilde{x}_{1,k+1}+\widehat{M}^{-1}(x_{1,k})\left( \widetilde{V}_{k}^{T}\varrho +\in _{k}\right) \beta _{k}\right] ^{T}\right) \nonumber \\&\quad =-\widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}+B\widetilde{x}_{1,k+1} \widetilde{x}_{1,k+1}^{T}B^{T}\nonumber \\&\qquad +2\widehat{M}^{-1}(x_{1,k})\widetilde{V}_{k}^{T}\varrho \beta _{k}\widetilde{x}_{1,k+1}^{T}B^{T}\nonumber \\&\qquad +2\widehat{M}^{-1}(x_{1,k})\in _{k}\beta _{k}\widetilde{x}_{1,k+1}^{T}B^{T}\nonumber \\&\qquad +\widehat{M}^{-1}(x_{1,k})\widetilde{V}_{k}^{T}\varrho \beta _{k}\beta _{k} ^{T}\varrho ^{T}\widetilde{V}_{k}\widehat{M}^{-T}(x_{1,k}) \nonumber \\&\qquad +2\widehat{M}^{-1}(x_{1,k})\widetilde{V}_{k}^{T}\varrho \beta _{k}\beta _{k} ^{T}\in _{k}^{T}\widehat{M}^{-T}(x_{1,k})\nonumber \\&\qquad +\widehat{M}^{-1}(x_{1,k})\in _{k}\beta _{k}\beta _{k}^{T}\in _{k}^{T}\widehat{M}^{-T}(x_{1,k}) \end{aligned}$$

(28)

Using the following inequalities:

$$\begin{aligned}&2\widehat{M}^{-1}(x_{1,k})\widetilde{V}_{k}^{T}\varrho \beta _{k}\widetilde{x}_{1,k+1}^{T}B^{T}\nonumber \\&\quad \le B\widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T} B^{T}\nonumber \\&\qquad +\widehat{M}^{-1}(x_{1,k})\widetilde{V}_{k}^{T}\varrho \beta _{k}\beta _{k} ^{T}\varrho ^{T}\widetilde{V}_{k}\widehat{M}^{-T}(x_{1,k}) \end{aligned}$$

(29)

$$\begin{aligned}&2\widehat{M}^{-1}(x_{1,k})\in _{k}\beta _{k}\widetilde{x}_{1,k+1}^{T}B^{T}\nonumber \\&\quad \le B\widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}B^{T}\nonumber \\&\qquad +\widehat{M}^{-1}(x_{1,k})\in _{k}\beta _{k}\beta _{k}^{T}\in _{k}^{T}\widehat{M}^{-T}(x_{1,k}) \end{aligned}$$

(30)

$$\begin{aligned}&2\widehat{M}^{-1}(x_{1,k})\widetilde{V}_{k}^{T}\varrho \beta _{k}\beta _{k} ^{T}\in _{k}^{T}\widehat{M}^{-T}(x_{1,k}) \nonumber \\&\quad \le \widehat{M}^{-1}(x_{1,k})\in _{k}\beta _{k}\beta _{k}^{T}\in _{k}^{T} \widehat{M}^{-T}(x_{1,k}) \nonumber \\&\qquad +\widehat{M}^{-1}(x_{1,k})\widetilde{V}_{k}^{T}\varrho \beta _{k}\beta _{k} ^{T}\varrho ^{T}\widetilde{V}_{k}\widehat{M}^{-T}(x_{1,k}) \end{aligned}$$

(31)

Substituting (29), (30), and (31) into (28) gives:

$$\begin{aligned}&\widetilde{x}_{1,k+2}\widetilde{x}_{1,k+2}^{T}-\widetilde{x}_{1,k+1} \widetilde{x}_{1,k+1}^{T}\nonumber \\&\quad \le -\widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}+3B\widetilde{x} _{1,k+1}\widetilde{x}_{1,k+1}^{T}B^{T}\nonumber \\&\qquad +3\widehat{M}^{-1}(x_{1,k})\widetilde{V}_{k}^{T}\varrho \beta _{k}\beta _{k} ^{T}\varrho ^{T}\widetilde{V}_{k}\widehat{M}^{-T}(x_{1,k})\nonumber \\&\qquad +3\widehat{M}^{-1}(x_{1,k})\in _{k}\beta _{k}\beta _{k}^{T}\in _{k}^{T}\widehat{M}^{-T}(x_{1,k}) \end{aligned}$$

(32)

(32) can be rewritten as follows:

$$\begin{aligned}&tr\left\{ \widetilde{x}_{1,k+2}\widetilde{x}_{1,k+2}^{T}\right\} -tr\left\{ \widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}\right\} \nonumber \\&\quad \le -\left[ 1-3\lambda _{\max }^{2}\left( B\right) \right] tr\left\{ \widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}\right\} \nonumber \\&\qquad +3tr\left\{ \widehat{M}^{-1}(x_{1,k})\widetilde{V}_{k}^{T}\varrho \beta _{k}\beta _{k}^{T}\varrho ^{T}\widetilde{V}_{k}\widehat{M}^{-T}(x_{1,k})\right\} \nonumber \\&\qquad +3tr\left\{ \widehat{M}^{-1}(x_{1,k})\in _{k}\beta _{k}\beta _{k}^{T}\in _{k} ^{T}\widehat{M}^{-T}(x_{1,k})\right\} \end{aligned}$$

(33)

Using (15), (16), and (18) in (33) gives:

$$\begin{aligned}&tr\left\{ \widetilde{x}_{1,k+2}\widetilde{x}_{1,k+2}^{T}\right\} -tr\left\{ \widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}\right\} \nonumber \\&\quad \le -\left[ 1-3\lambda _{\max }^{2}\left( B\right) \right] tr\left\{ \widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}\right\} \nonumber \\&\qquad +3\gamma _{\max }^{2}\beta _{\max }^{2}tr\left\{ \widetilde{V}_{k}^{T} \varrho \varrho ^{T}\widetilde{V}_{k}\right\} \nonumber \\&\qquad +3\gamma _{\max }^{2}\beta _{\max }^{2}tr\left\{ \overline{\in }\overline{\in } ^{T}\right\} \end{aligned}$$

(34)

Using the following equality:

$$\begin{aligned}&atr\left\{ \widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}\right\} +2atr\left\{ \widetilde{V}_{k}^{T}\varrho \widetilde{x}_{1,k+1}^{T}\right\} \nonumber \\&\qquad +atr\left\{ \widetilde{V}_{k}^{T}\varrho \varrho ^{T}\widetilde{V}_{k}\right\} \nonumber \\&\qquad -a\left[ \widetilde{x}_{1,k+1}+\widetilde{V}_{k}^{T}\varrho \right] \left[ \widetilde{x}_{1,k+1}+\widetilde{V}_{k}^{T}\varrho \right] ^{T}=0 \end{aligned}$$

(35)

where $a>0\in \mathfrak {R}$. Substituting (35) into (34) gives:

$$\begin{aligned}&tr\left\{ \widetilde{x}_{1,k+2}\widetilde{x}_{1,k+2}^{T}\right\} -tr\left\{ \widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}\right\} \nonumber \\&\quad \le -\left[ 1-3\lambda _{\max }^{2}\left( B\right) -a\right] tr\left\{ \widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}\right\} \nonumber \\&\qquad +\left[ 3\gamma _{\max }^{2}\beta _{\max }^{2}+a\right] tr\left\{ \widetilde{V}_{k}^{T}\varrho \varrho ^{T}\widetilde{V}_{k}\right\} \nonumber \\&\qquad +2atr\left\{ \widetilde{V}_{k}^{T}\varrho \widetilde{x}_{1,k+1}^{T}\right\} +3\gamma _{\max }^{2}\beta _{\max }^{2}tr\left\{ \overline{\in }\overline{\in } ^{T}\right\} \nonumber \\&\qquad -\,a\left[ \widetilde{x}_{1,k+1}+\widetilde{V}_{k}^{T}\varrho \right] \left[ \widetilde{x}_{1,k+1}+\widetilde{V}_{k}^{T}\varrho \right] ^{T} \end{aligned}$$

(36)

Using the definitions of $\alpha$, $b$, and $c$ of (18), (36) can be written as:

$$\begin{aligned}&tr\left\{ \widetilde{x}_{1,k+2}\widetilde{x}_{1,k+2}^{T}\right\} -tr\left\{ \widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}\right\} \nonumber \\&\quad \le -\,\alpha tr\left\{ \widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1}^{T}\right\} +btr\left\{ \widetilde{V}_{k}^{T}\varrho \varrho ^{T}\widetilde{V}_{k}\right\} \nonumber \\&\qquad +2atr\left\{ \widetilde{V}_{k}^{T}\varrho \widetilde{x}_{1,k+1}^{T}\right\} +ctr\left\{ \overline{\in }\overline{\in }^{T}\right\} \end{aligned}$$

(37)

Substituting (37) into (27) gives:

$$\begin{aligned} \Delta L_{k+1}&\le -\frac{\alpha }{2}tr\left\{ \widetilde{x}_{1,k+1} \widetilde{x}_{1,k+1}^{T}\right\} +\frac{c}{2}tr\left\{ \overline{\in }\overline{\in }^{T}\right\} \nonumber \\&\quad +\frac{b}{2}tr\left\{ \widetilde{V}_{k}^{T}\varrho \varrho ^{T}\widetilde{V}_{k}\right\} +atr\left\{ \widetilde{V}_{k}^{T}\varrho \widetilde{x}_{1,k+1}^{T}\right\} \nonumber \\&\quad -\frac{1}{2}tr\left\{ \left( \widetilde{V}_{k}-\widetilde{V}_{k-1}\right) ^{T}P_{k-1}^{-1}\left( \widetilde{V}_{k}-\widetilde{V}_{k-1}\right) \right\} \nonumber \\&\quad +\frac{1}{2}tr\left\{ \widetilde{V}_{k}^{T}\left( P_{k}^{-1}-P_{k-1} ^{-1}\right) \widetilde{V}_{k}\right\} \nonumber \\&\quad +tr\left\{ \widetilde{V}_{k}^{T}P_{k-1}^{-1}\left( \widetilde{V} _{k}-\widetilde{V}_{k-1}\right) \right\} \nonumber \\&\le -\frac{\alpha }{2}tr\left\{ \widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1} ^{T}\right\} +\frac{c}{2}tr\left\{ \overline{\in }\overline{\in }^{T}\right\} \nonumber \\&\quad -\frac{1}{2}tr\left\{ \left( \widetilde{V}_{k}-\widetilde{V}_{k-1}\right) ^{T}P_{k-1}^{-1}\left( \widetilde{V}_{k}-\widetilde{V}_{k-1}\right) \right\} \nonumber \\&\quad +\frac{1}{2}tr\left\{ \widetilde{V}_{k}^{T}\left( P_{k}^{-1}-P_{k-1} ^{-1}-b\varrho \varrho ^{T}\right) \widetilde{V}_{k}\right\} \nonumber \\&\quad +tr\left\{ \widetilde{V}_{k}^{T}P_{k-1}^{-1}\right. \nonumber \\&\quad *\left. \left( \widetilde{V}_{k}-\widetilde{V}_{k-1}+P_{k-1}\left\{ a\varrho \widetilde{x}_{1,k+1}^{T}+b\varrho \varrho ^{T}\widetilde{V} _{k}\right\} \right) \right\} \nonumber \\ \end{aligned}$$

(38)

The term $P_{k}^{-1}-P_{k-1}^{-1}-b\varrho \varrho ^{T}$ of (38) may be equal to zero to reach the objective $\Delta L_{k+1}\le 0$; therefore, using the matrix inversion lemma gives:

$$\begin{aligned}&P_{k}^{-1}=P_{k-1}^{-1}+b\varrho \varrho ^{T} \nonumber \\&\quad \Longrightarrow P_{k}=\left[ P_{k-1}^{-1}+b\varrho \varrho ^{T}\right] ^{-1}\nonumber \\&\quad \Longrightarrow P_{k}=P_{k-1}-\left[ \frac{1}{b}I+\varrho \varrho ^{T} P_{k-1}\right] ^{-1}P_{k-1}\varrho \varrho ^{T}P_{k-1}\nonumber \\ \end{aligned}$$

(39)

This gives (11). The term $\widetilde{V}_{k}-\widetilde{V} _{k-1}+P_{k-1}\left\{ a\varrho \widetilde{x}_{1,k+1}^{T}\right.$ $\left. +b\varrho \varrho ^{T}\widetilde{V}_{k}\right\}$ may be equal to zero to reach the objective $\Delta L_{k+1}\le 0$ which gives:

$$\begin{aligned}&\widetilde{V}_{k}-\widetilde{V}_{k-1}+P_{k-1}\left\{ a\varrho \widetilde{x}_{1,k+1}^{T}+b\varrho \varrho ^{T}\widetilde{V}_{k}\right\} =0\nonumber \\&\quad \Longrightarrow \left[ I+bP_{k-1}\varrho \varrho ^{T}\right] \widetilde{V} _{k}=\widetilde{V}_{k-1}-P_{k-1}\left\{ a\varrho \widetilde{x}_{1,k+1} ^{T}\right\} \nonumber \\&\quad \Longrightarrow \left[ I+bP_{k-1}\varrho \varrho ^{T}\right] \widetilde{V} _{k}=\left[ I+bP_{k-1}\varrho \varrho ^{T}\right] \widetilde{V}_{k-1}\nonumber \\&\quad P_{k-1}\left\{ a\varrho \widetilde{x}_{1,k+1}^{T}+b\varrho \varrho ^{T}\widetilde{V}_{k-1}\right\} \nonumber \\&\quad \Longrightarrow \widetilde{V}_{k}=\widetilde{V}_{k-1} \nonumber \\&\qquad -\left[ I+bP_{k-1}\varrho \varrho ^{T}\right] ^{-1}P_{k-1}\left\{ a\varrho \widetilde{x}_{1,k+1}^{T}+b\varrho \varrho ^{T}\widetilde{V} _{k-1}\right\} \nonumber \\ \end{aligned}$$

(40)

Thus (11) is derived. From (38) finally gives:

$$\begin{aligned} \Delta L_{k+1}&\le -\frac{\alpha }{2}tr\left\{ \widetilde{x}_{1,k+1} \widetilde{x}_{1,k+1}^{T}\right\} +\frac{c}{2}tr\left\{ \overline{\in }\overline{\in }^{T}\right\} \nonumber \\&-\frac{1}{2}tr\left\{ \left( \widetilde{V}_{k}-\widetilde{V}_{k-1}\right) ^{T}P_{k-1}^{-1}\left( \widetilde{V}_{k}-\widetilde{V}_{k-1}\right) \right\} \end{aligned}$$

(41)

From Rubio et al. (2011), Rubio (2012), and using (26), (41), it is guaranteed that the tracking error and parameters error of the closed-loop system are uniform stable and $L_{k}$ is bounded. Considering the addition of (41) from $1$ to $T$ and using that $L_{T}>0$ and that $L_{1}$ is a constant:

$$\begin{aligned}&L_{T}-L_{1}\le - {\displaystyle \sum \limits _{k=1}^{T}} \frac{\alpha }{2}tr\left\{ \widetilde{x}_{1,k+1}\widetilde{x}_{1,k+1} ^{T}\right\} +T\frac{c}{2}tr\left\{ \overline{\in }\overline{\in }^{T}\right\} \\&\qquad -{\displaystyle \sum \limits _{k=1}^{T}} \frac{1}{2}tr\left\{ \left( \widetilde{V}_{k}-\widetilde{V}_{k-1}\right) ^{T}P_{k-1}^{-1}\left( \widetilde{V}_{k}-\widetilde{V}_{k-1}\right) \right\} \\&\quad \Longrightarrow \underset{T\rightarrow \infty }{\lim }\frac{1}{T} {\displaystyle \sum \limits _{k=1}^{T}} \left[ \frac{1}{2}tr\left\{ \left( \widetilde{V}_{k}-\widetilde{V} _{k-1}\right) ^{T}P_{k-1}^{-1}\left( \widetilde{V}_{k}-\widetilde{V} _{k-1}\right) \right\} \right. \\&\qquad \left. +\frac{\alpha }{2}tr\left\{ \widetilde{x}_{1,k+1}\widetilde{x} _{1,k+1}^{T}\right\} \right] \le L_{1}\underset{T\rightarrow \infty }{\lim }\frac{1}{T}+\frac{c}{2}tr\left\{ \overline{\in }\overline{\in }^{T}\right\} \end{aligned}$$

Thus (18) is satisfied.