Decentralized robust tracking control for uncertain robots

Abstract

To realize decentralized robust tracking control for robots with uncertain parameters, a new design method is proposed. A robust tracking controller designed by this method consists of two parts: a feedforward controller and a feedback robust controller. A feedforward control is first applied and error dynamics are introduced. For each joint error subsystem a robust controller is designed in two steps: first, a nominal controller is designed for the nominal plant to achieve desired tracking performance, then a robust compensator is added to restrain the influence of the perturbation, that is the difference of the real plant from the nominal plant. The controller designed by the proposed method is a linear time-invariant one. It is shown that robust stability and robust tracking property can be achieved by applying the controller with a sufficiently wide frequency bandwidth. An important feature of the method is that the controller parameters can be tuned on-line easily.

Appendices

Appendix 1: Proof of Lemma 1

Solving Eq. 17, one can find that ψ _i =P _i B _ql can be expressed in the form as stated in the lemma, and under the condition that $\frac{1} {{p_{i} }} > > f \geqslant 1$, γ _ji can be represented as:

$$ \gamma _{{ji}} = \frac{{a_{{2i}} f^{2}_{i} + a_{{1i}} f_{i} + a_{{0i}} }} {{f^{2}_{i} + (\alpha _{{1i}} + \beta _{{1i}} )f_{i} + (\alpha _{{1i}} \beta _{{1i}} + \beta _{{2i}} ) + (\alpha _{{1i}} \beta _{{2i}} )f^{{ - 1}}_{i} }},\quad j = 1,2,3 $$

$$ \gamma _{{ji}} = \frac{{b_{{3i}} f^{3}_{i} + b_{{2i}} f^{2}_{i} + b_{{1i}} f_{i} + b_{{0i}} }} {{f^{3}_{i} + (\alpha _{{1i}} + \beta _{{1i}} )f^{2}_{i} + (\alpha _{{1i}} \beta _{{1i}} + \beta _{{2i}} )f_{i} + \alpha _{{1i}} \beta _{{2i}} }},\quad j = 4,5 $$

where a _ji (j=0,1,2) and b _ki (k=0,1,2,3) are constants independent of f _i . Since α _1i ,β _1i ,β _2i >0, obviously, the denominator of γ _ji is larger than zero, so γ _ji is continuous in f _i and can be bounded by constants that are independent of f _i . □

Appendix 2: Proof of Lemma 2

From (5), we have

$$ \begin{aligned} {\left\| \Delta \right\|} \leqslant {\left\| {M(\theta ) - M(\theta _{{\text{d}}} )} \right\|}{\left\| {\ddot{\theta }_{{\text{d}}} } \right\|} + {\left\| {M(\theta _{{\text{d}}} )\ddot{\theta }_{{\text{d}}} - M_{0} (\theta _{{\text{d}}} )\ddot{\theta }_{{\text{d}}} } \right\|} & \\ + {\left\| {C(\theta ,\dot{\theta }) - C(\theta _{{\text{d}}} ,\dot{\theta }_{{\text{d}}} )} \right\|}{\left\| {\dot{\theta }_{{\text{d}}} } \right\|} + {\left\| {C(\theta _{{\text{d}}} ,\dot{\theta }_{{\text{d}}} )\dot{\theta }_{{\text{d}}} - C_{0} (\theta _{{\text{d}}} ,\dot{\theta }_{{\text{d}}} )\dot{\theta }_{{\text{d}}} } \right\|} & \\ + {\left\| {g(\theta ) - g(\theta _{{\text{d}}} )} \right\|} + {\left\| {g(\theta _{{\text{d}}} ) - g_{0} (\theta _{{\text{d}}} )} \right\|} & \\ \end{aligned} $$

Suppose the desired trajectory is bounded and satisfies

$$ {\left\| {\theta _{d} } \right\|} \leqslant c_{1} ,{\left\| {\ifmmode\expandafter\dot\else\expandafter\.\fi{\theta }_{d} } \right\|} \leqslant c_{2} ,{\left\| {\ifmmode\expandafter\ddot\else\expandafter\"\fi{\theta }_{d} } \right\|} \leqslant c_{3} $$

where c ₁, c ₂, and c ₃ are constants. In view of Properties 1–3 in Sect. 2, there exist positive constants k _M, k _M1, k _c, k _c1, k _c2 satisfying

$$ {\left\| \Delta \right\|} \leqslant k_{{\text{M}}} c_{3} + k_{{{\text{M}}1}} c_{3} {\left\| e \right\|} + k_{{\text{C}}} c_{2} + k_{{{\text{C}}1}} c_{2} {\left\| {\dot{e}} \right\|} + k_{{{\text{C}}2}} c^{2}_{2} {\left\| e \right\|} + k_{{\text{g}}} + k_{{{\text{g}}1}} {\left\| e \right\|} $$

From Property 1 and Property 3(3), there exist positive constants k _c3, k _M2 such that

$$ {\left\| { - M^{{ - 1}} C\dot{e} + (M^{{ - 1}} - b_{0} )u_{0} } \right\|} \leqslant k_{{{\text{C}}3}} {\left\| {\dot{e}} \right\|}^{2} + k_{{{\text{M}}2}} {\left\| {u_{0} } \right\|} $$

So we have

$$ {\left\| {\tilde{q}} \right\|} \leqslant k_{{{\text{q}}1}} + k_{{{\text{q}}2}} {\left\| e \right\|} + k_{{{\text{q}}3}} {\left\| {\dot{e}} \right\|} + k_{{{\text{q}}4}} {\left\| {\dot{e}} \right\|}^{2} + k_{{{\text{q}}5}} {\left\| {u_{0} } \right\|}^{2} $$

where k _q1, k _q2, ..., k _q5 are bounded positive constants. □

Appendix 3: Proof of Lemma 3

Because b ₀≥λ _min(M ⁻¹), which implies that $\bar{\delta }(b^{{ - 1}}_{0} M^{{ - 1}} ) \leqslant I$, we have

$$ \bar{\delta }[b^{{ - 1}}_{0} M^{{ - 1}} - I] \leqslant 1 - b^{{ - 1}}_{0} \lambda _{{\min }} (M^{{ - 1}} ) $$

From Assumption (C), it follows that

$$ \ifmmode\expandafter\bar\else\expandafter\=\fi{\delta }[b^{{ - 1}}_{0} M^{{ - 1}} - I] \leqslant 1 - \varepsilon _{\Delta } $$

Therefore the conclusion holds. □

Appendix 4: Proof of Lemma 4

From (20), if $f_{i} \geqslant {\mathop {\max }\limits_{j = 1}^3 }(\gamma _{{j\max }} )^{2}$, then

$$ {\text{ }}\frac{{k_{{{\text{q}}1}} }} {{b_{0} {\sqrt {f_{i} } }}}{\sum\limits_{i = 1}^3 {\gamma _{{i\max }} } } \leqslant \frac{{3k_{{{\text{q}}1}} }} {{b_{0} }} $$

(21)

Meanwhile, if

$$ \mu \geqslant \gamma _{{4\max }} + \gamma _{{5\max }} $$

which is satisfied under Assumption (A), then we have

$$ \frac{{k_{{{\text{q}}1}} }} {{b_{0} }}[1 + {(\gamma _{{4\max }} + \gamma _{{5\max }} )} \mathord{\left/ {\vphantom {{(\gamma _{{4\max }} + \gamma _{{5\max }} )} \mu }} \right. \kern-\nulldelimiterspace} \mu ] \leqslant \frac{{2k_{{{\text{q}}1}} }} {{b_{0} }} $$

(22)

Combining (21) and (22) results in

$$ \mu _{0} \leqslant \frac{{5k_{{{\text{q}}1}} }} {{b_{0} }} $$

□

Appendix 5: Proof of Theorem 1

Consider the following Lyapunov function candidate

$$ V = X^{T} PX = {\sum\limits_{i = 1}^n {V_{i} } },\quad V_{i} = X^{T}_{i} P_{i} X_{i} $$

where

$$ X = {\left[ {X^{T}_{1} ,X^{T}_{2} , \cdots ,X^{T}_{n} } \right]}^{T} ,\quad \;P = {\left[ {\begin{array}{*{20}c} {{P_{1} }} & {{}} & {{}} \\ {{}} & { \ddots } & {{}} \\ {{}} & {{}} & {{P_{n} }} \\ \end{array} } \right]} $$

(23)

The derivative of V, along the solution of the controlled system (16), is given by

$$ \begin{array}{*{20}l} {{\dot{V}} \hfill} & { = \hfill} & {{{\sum\limits_{i = 1}^n {\dot{V}_{i} } } = {\sum\limits_{i = 1}^n {( - X^{T}_{i} X_{i} + 2X^{T}_{i} P_{i} B_{{{\text{q}}i}} q_{i} )} }} \hfill} \\ {{} \hfill} & { \leqslant \hfill} & {{ - (e^{T} e + \dot{e}^{T} \dot{e} + u^{T}_{0} u_{0} + v^{T}_{0} v_{0} + \tilde{v}^{T} \tilde{v}) + 2\left\{ {{\sum\limits_{i = 1}^n {\frac{1} {{f_{i} b_{0} }}} }\gamma _{{1i}} e_{i} q_{i} } \right.} \hfill} \\ {{} \hfill} & {{} \hfill} & {\begin{aligned} & + {\sum\limits_{i = 1}^n {\frac{1} {{f_{i} b_{0} }}} }\gamma _{{2i}} \dot{e}_{i} q_{i} + {\sum\limits_{i = 1}^n {\frac{1} {{f_{i} b_{0} }}} }\gamma _{{3i}} u_{{0i}} q_{i} \\ & + {\sum\limits_{i = 1}^n {\frac{1} {{{\sqrt {f_{i} } }b_{0} }}(1 + \frac{{\gamma _{{4i}} }} {\mu })v_{{0i}} q_{i} + \left. {{\sum\limits_{i = 1}^n {\frac{{\gamma _{{5i}} }} {{{\sqrt {f_{i} } }b_{0} \mu }}\tilde{v}_{i} q_{i} } }} \right\}} } \\ \end{aligned} \hfill} \\ {{} \hfill} & { = \hfill} & {{ - (e^{T} e + \dot{e}^{T} \dot{e} + u^{T}_{0} u_{0} + v^{T}_{0} v_{0} + \tilde{v}^{T} \tilde{v})} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + 2\left[ {e^{T} {\text{diag}}{\left\{ {\frac{{\gamma _{{1i}} }} {{f_{i} b_{0} }}} \right\}} + \dot{e}^{T} {\text{diag}}{\left\{ {\frac{{\gamma _{{2i}} }} {{f_{i} b_{0} }}} \right\}} + u^{T}_{0} {\text{diag}}{\left\{ {\frac{{\gamma _{{3i}} }} {{f_{i} b_{0} }}} \right\}}} \right.} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + v^{T}_{0} {\text{diag}}{\left\{ {\frac{1} {{{\sqrt {f_{i} } }b_{0} }}(1 + \frac{{\gamma _{{4i}} }} {\mu })} \right\}} + \left. {\tilde{v}^{T} {\text{diag}}{\left\{ {\frac{{\gamma _{{5i}} }} {{{\sqrt {f_{i} } }b_{0} \mu }}} \right\}}} \right]q} \hfill} \\ \end{array} $$

(24)

From Lemma 2 and the definition of γ _jmax in Lemma 4, (24) becomes

$$ \begin{array}{*{20}l} {{\dot{V}} \hfill} & { \leqslant \hfill} & {{ - ({\left\| e \right\|}^{2} + {\left\| {\dot{e}} \right\|}^{2} + {\left\| {u_{0} } \right\|}^{2} + {\left\| {v_{0} } \right\|}^{2} + {\left\| {\tilde{v}} \right\|}^{2} )} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + 2\left\{ {\left[ {\frac{1} {{f_{i} b_{0} }}} \right.} \right.(\gamma _{{1\max }} {\left\| e \right\|} + \gamma _{{2\max }} {\left\| {\dot{e}} \right\|} + \gamma _{{3\max }} {\left\| {u_{0} } \right\|})} \hfill} \\ {{} \hfill} & {{} \hfill} & {{\left. { + \frac{1} {{{\sqrt {f_{i} } }b_{0} }}(1 + \frac{{\gamma _{{4\max }} }} {\mu }){\left\| {v_{0} } \right\|} + \frac{{\gamma _{{5\max }} }} {{b_{0} \mu }}{\left\| {\tilde{v}} \right\|}} \right] \cdot {\left\| {\tilde{q}} \right\|}} \hfill} \\ {{} \hfill} & {{} \hfill} & {\begin{aligned} & + \left[ {\frac{1} {{{\sqrt {f_{i} } }}}} \right.(\gamma _{{1\max }} {\left\| e \right\|} + \gamma _{{2\max }} {\left\| {\dot{e}} \right\|} + \gamma _{{3\max }} {\left\| {u_{0} } \right\|}) \\ & + \left. {(1 + \frac{{\gamma _{{4\max }} }} {\mu }){\left\| {v_{0} } \right\|} + \left. {\frac{{\gamma _{{5\max }} }} {\mu }{\left\| {\tilde{v}} \right\|}} \right] \cdot {\left\| {b^{{ - 1}}_{0} (M^{{ - 1}} - b_{0} I)} \right\|} \cdot {\left\| {\tilde{v}} \right\|}} \right\} \\ \end{aligned} \hfill} \\ \end{array} $$

(25)

In the view of Lemma 1 and Lemma 3, (25) becomes

$$ \begin{array}{*{20}l} {{\ifmmode\expandafter\dot\else\expandafter\.\fi{V}} \hfill} & { \leqslant \hfill} & {{ - ({\left\| e \right\|}^{2} + {\left\| {\ifmmode\expandafter\dot\else\expandafter\.\fi{e}} \right\|}^{2} + {\left\| {u_{0} } \right\|}^{2} + {\left\| {v_{0} } \right\|}^{2} + {\left\| { \ifmmode\expandafter\tilde\else\expandafter\~\fi{v}} \right\|}^{2} )} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + 2\left\{ {[\frac{1} {{f_{i} b_{0} }}(\gamma _{{1\max }} {\left\| e \right\|} + \gamma _{{2\max }} {\left\| {\ifmmode\expandafter\dot\else\expandafter\.\fi{e}} \right\|} + \gamma _{{3\max }} {\left\| {u_{0} } \right\|})} \right.} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + \frac{1} {{{\sqrt {f_{i} } }b_{0} }}(1 + \frac{{\gamma _{{4\max }} }} {\mu }){\left\| {v_{0} } \right\|} + \frac{{\gamma _{{5\max }} }} {{b_{0} \mu }}{\left\| { \ifmmode\expandafter\tilde\else\expandafter\~\fi{v}} \right\|}]} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ \cdot [k_{{{\text{q}}1}} + k_{{{\text{q}}2}} {\left\| e \right\|} + k_{{{\text{q}}3}} {\left\| {\ifmmode\expandafter\dot\else\expandafter\.\fi{e}} \right\|} + k_{{{\text{q}}4}} {\left\| {\ifmmode\expandafter\dot\else\expandafter\.\fi{e}} \right\|}^{2} + k_{{{\text{q}}5}} {\left\| {u_{0} } \right\|}]} \hfill} \\ {{} \hfill} & {{} \hfill} & {\begin{aligned} & + \frac{1} {{{\sqrt {f_{i} } }}}(\gamma _{{1\max }} {\left\| e \right\|} + \gamma _{{2\max }} {\left\| {\ifmmode\expandafter\dot\else\expandafter\.\fi{e}} \right\|} + \gamma _{{3\max }} {\left\| {u_{0} } \right\|}){\left\| { \ifmmode\expandafter\tilde\else\expandafter\~\fi{v}} \right\|} \\ & + \left. {\ifmmode\expandafter\bar\else\expandafter\=\fi{\delta }(b^{{ - 1}}_{0} (M^{{ - 1}} - b_{0} I))[(1 + \frac{{\gamma _{{4\max }} }} {\mu }){\left\| {v_{0} } \right\|} + \frac{{\gamma _{{5\max }} }} {\mu }{\left\| { \ifmmode\expandafter\tilde\else\expandafter\~\fi{v}} \right\|}] \cdot {\left\| { \ifmmode\expandafter\tilde\else\expandafter\~\fi{v}} \right\|}} \right\} \\ \end{aligned} \hfill} \\ \end{array} $$

(26)

Define k _sum=k _q1+k _q2+k _q3+k _q5 and k=[k _q2, k _q3, k _q5]^T and let k _i denote the ith element of the vector k. Then (26) is further expressed as

$$ \dot{V} \leqslant - \mu _{1} {\left\| e \right\|}^{2} - \mu _{2} {\left\| {\dot{e}} \right\|}^{2} - \mu _{3} {\left\| {u_{0} } \right\|}^{2} - \mu _{4} {\left\| {v_{0} } \right\|}^{2} - \mu _{5} {\left\| {\tilde{v}} \right\|}^{2} + \frac{{\mu _{0} }} {{{\sqrt {f_{i} } }}} $$

(27)

where μ ₀ is defined as (20),

$$ \mu _{i} = \left\{ {\begin{array}{*{20}l} {{1 - {\rho _{i} } \mathord{\left/ {\vphantom {{\rho _{i} } {{\sqrt {f_{i} } }}}} \right. \kern-\nulldelimiterspace} {{\sqrt {f_{i} } }}} \hfill} & {{(i = 1,3)} \hfill} \\ {{1 - ( \ifmmode\expandafter\tilde\else\expandafter\~\fi{\rho }_{i} + {\rho _{i} )} \mathord{\left/ {\vphantom {{\rho _{i} )} {{\sqrt {f_{i} } }}}} \right. \kern-\nulldelimiterspace} {{\sqrt {f_{i} } }}} \hfill} & {{(i = 2)} \hfill} \\ {{1 - \ifmmode\expandafter\bar\else\expandafter\=\fi{\delta }(b^{{ - 1}}_{0} (M^{{ - 1}} - b_{0} I))(1 + {\gamma _{{4\max }} } \mathord{\left/ {\vphantom {{\gamma _{{4\max }} } \mu }} \right. \kern-\nulldelimiterspace} \mu ) - {\rho _{i} } \mathord{\left/ {\vphantom {{\rho _{i} } {{\sqrt {f_{i} } }}}} \right. \kern-\nulldelimiterspace} {{\sqrt {f_{i} } }}} \hfill} & {{(i = 4)} \hfill} \\ {{1 - \ifmmode\expandafter\bar\else\expandafter\=\fi{\delta }(b^{{ - 1}}_{0} (M^{{ - 1}} - b_{0} I))[1 + {(\gamma _{{4\max }} + 2\gamma _{{5\max }} )} \mathord{\left/ {\vphantom {{(\gamma _{{4\max }} + 2\gamma _{{5\max }} )} \mu }} \right. \kern-\nulldelimiterspace} \mu ] - {\rho _{i} } \mathord{\left/ {\vphantom {{\rho _{i} } {{\sqrt {f_{i} } }}}} \right. \kern-\nulldelimiterspace} {{\sqrt {f_{i} } }}} \hfill} & {{(i = 5)} \hfill} \\ \end{array} } \right. $$

(28)

$$ \rho _{i} = \left\{ {\begin{array}{*{20}c} {\frac{1} {{{\sqrt {f_{i} } }b_{0} }}[\gamma _{{i\max }} k_{{{\text{sum}}}} + k_{i} {\sum\limits_{j = 1}^3 {\gamma _{{j\max }} } }] + \frac{{k_{i} }} {{b_{0} }}[1 + \frac{{\gamma _{{4\max }} + \gamma _{{5\max }} }} {\mu }] + \gamma _{{i\max }} }{(i = 1,2,3)} \\ {\frac{1} {{b_{0} }}[1 + \frac{{\gamma _{{4\max }} }} {\mu }]k_{{{\text{sum}}}} }{(i = 4)} \\ {{\sum\limits_{j = 1}^3 {\gamma _{{j\max }} } } + \frac{{\gamma _{{5\max }} k_{{{\text{sum}}}} }} {{b_{0} \mu }}}{(i = 5)} \\ \end{array} } \right. $$

(29)

$$ \begin{aligned} \tilde{\rho }_{2} = \frac{2} {{\lambda _{b} }}[\frac{{\gamma _{{1\max }} }} {{{\sqrt {f_{i} } }}}{\left\| e \right\|} + \frac{{\gamma _{{2\max }} }} {{{\sqrt {f_{i} } }}}{\left\| {\dot{e}} \right\|} + \frac{{\gamma _{{3\max }} }} {{{\sqrt {f_{i} } }}}{\left\| {u_{0} } \right\|} + (1 + \frac{{\gamma _{{4\max }} }} {\mu }){\left\| {v_{0} } \right\|} & \\ \quad \;\; + \frac{{\gamma _{{5\max }} }} {\mu }{\left\| {\tilde{v}} \right\|}]k_{{{\text{q}}4}} & \\ \end{aligned} $$

(30)

Define $Y = {\left[ {e^{T} ,\dot{e}^{T} ,u^{T}_{0} ,v^{T}_{0} ,\tilde{v}^{T} } \right]}^{T} = {\left[ {Y^{T}_{1} ,Y^{T}_{2} ,Y^{T}_{3} ,Y^{T}_{4} ,Y^{T}_{5} } \right]}^{T}$, then under Assumption (A), we have

$$ \ifmmode\expandafter\tilde\else\expandafter\~\fi{\rho }_{2} \leqslant \phi _{1} {\sum\limits_{j = 1}^5 {{\left\| {Y_{i} } \right\|}} } \leqslant {\sqrt 5 }\phi _{1} {\left\| Y \right\|} $$

(31)

where $\phi _{1} = 2k_{{{\text{q}}4}} b^{{ - 1}}_{0} {\mathop {\max }\limits_{} }\,\{ \gamma _{{1\max }} ,\gamma _{{2\max }} ,\gamma _{{3\max }} ,2\}$, is a bounded constant independent of f _i .

In the following, the conditions are analyzed that ensure μ _i >0, i=1,2,...,5.

Let π _min be a positive constant so that π _min<min{ε _Δ,1}, and let p̄ _i denote the values of p _i when f _i =1 and μ=γ _4max+2γ _5max. Then under Assumption (A) and the condition that μ≥γ _4max+2γ _5max, we have p _i ≤p̄ _i .

If the following equalities hold

$$ f_{i} \geqslant f_{{{\text{c}}2}} ,{\text{ }}\mu \geqslant \gamma _{{4\max }} + 2\gamma _{{5\max }} $$

(32)

$$ {\left\| Y \right\|} \leqslant {\text{ }}\frac{1} {{{\sqrt {\text{5}} }\phi _{1} }}[(1 - \pi _{{\min }} ){\sqrt {f_{i} } } - \ifmmode\expandafter\bar\else\expandafter\=\fi{\rho }_{2} ] $$

(33)

where

$$ f_{{{\text{c}}2}} = \max {\left\{ {{\mathop {\max }\limits_{i = 1,3} }{\left( {\frac{{\bar{\rho }_{i} }} {{1 - \pi _{{\min }} }}} \right)}^{2} ,\;{\mathop {\max }\limits_{i = 4,5} }{\left( {\frac{{\bar{\rho }_{i} }} {{\varepsilon _{\Delta } - \pi _{{\min }} }}} \right)}^{2} } \right\}} $$

(34)

then by (28) and from Lemma 3, we have

$$ \mu _{i} \geqslant \pi _{{\min }} ,{\text{ }}i = 1,2, \ldots ,5 $$

(35)

Note that (33) defines the attractive region.

If (35) holds and f _i ≥f _c1 from Lemma 4, one sees that (27) can be rewritten as

$$ \dot{V} \leqslant - \pi _{{\min }} {\left\| Y \right\|}^{2} + \frac{{5k_{{{\text{q}}1}} }} {{{\sqrt {f_{i} } }b_{0} }} $$

(36)

Since

$$ {\left\| Y \right\|}^{2} = {\left\| X \right\|}^{2} \geqslant \frac{{X^{T} PX}} {{\lambda _{{\max }} (P)}} = \frac{V} {{\lambda _{{\max }} (P)}} $$

we obtain

$$ \dot{V} \leqslant - \zeta _{{\text{s}}} V + \varepsilon _{{\text{s}}} $$

(37)

and

$$ V(t) \leqslant V(t_{0} ){\text{e}}^{{ - \zeta _{{\text{s}}} (t - t_{0} )}} + \frac{{\varepsilon _{{\text{s}}} }} {{\zeta _{{\text{s}}} }}(1 - {\text{e}}^{{ - \zeta _{{\text{s}}} (t - t_{0} )}} ) $$

(38)

where

$$ \zeta _{{\text{s}}} = \frac{{\pi _{{\min }} }} {{\lambda _{{\max }} (P)}},\quad \varepsilon _{{\text{s}}} = \frac{{5k_{{{\text{q}}1}} }} {{{\sqrt {f_{i} } }b_{0} }} $$

Let

$$ \bar{V} = \max \{ V(t_{0} ),\;5k_{{{\text{q}}1}} b^{{ - 1}}_{0} \zeta ^{{ - 1}}_{{\text{s}}} \} $$

Then under Assumption (A) and from (37) it is known that $V(t) \leqslant \ifmmode\expandafter\bar\else\expandafter\=\fi{V},\forall t \geqslant t_{0}$ and ${\mathop {\lim }\limits_{t \to \infty } }V(t) \leqslant \varepsilon _{{\text{s}}} \zeta ^{{ - 1}}_{{\text{s}}}$. Because

$$ {\left\| Y \right\|}^{2} = {\left\| X \right\|}^{2} \leqslant \frac{{X^{T} PX}} {{\lambda _{{\min }} (P)}} = \frac{V} {{\lambda _{{\min }} (P)}} $$

(39)

Y(t) (and X(t) starting from the attractive region given by (33)) can remain inside that region, if

$$ \frac{1} {{{\sqrt {\text{5}} }\phi _{1} }}[(1 - \pi _{{\min }} ){\sqrt {f_{i} } } - \ifmmode\expandafter\bar\else\expandafter\=\fi{\rho }_{2} ] \geqslant {\sqrt {\ifmmode\expandafter\bar\else\expandafter\=\fi{V}\lambda ^{{ - 1}}_{{\min }} (P)} } $$

(40)

By Assumption (A), (40) holds, if

$$ f_{i} \geqslant f_{{{\text{c}}3}} ,\quad f_{{{\text{c}}3}} = \frac{1} {{(1 - \pi _{{\min }} )^{2} }}{\left( {\phi _{1} {\sqrt {5\ifmmode\expandafter\bar\else\expandafter\=\fi{V}\lambda ^{{ - 1}}_{{\min }} (P)} } + \ifmmode\expandafter\bar\else\expandafter\=\fi{\rho }_{2} } \right)}^{2} $$

(41)

In addition, from (37) and (39) we know that the state Y convergences to the set given by

$$ \Omega _{{\text{s}}} = {\left\{ {Y\left| {{\left\| Y \right\|}^{2} \leqslant {\varepsilon _{{\text{s}}} } \mathord{\left/ {\vphantom {{\varepsilon _{{\text{s}}} } {(\zeta _{{\text{s}}} \lambda _{{\min }} (P))}}} \right. \kern-\nulldelimiterspace} {(\zeta _{{\text{s}}} \lambda _{{\min }} (P))}} \right.} \right\}} $$

at a rate not slower than exp(−ξ _s t/2).

By (39), if

$$ V(t) \leqslant \lambda _{{\min }} (P)\eta $$

then

$$ {\left\| X \right\|}^{2} \leqslant \eta $$

When both of the closed-loop system and the reference model are of zero initial conditions, V(t ₀)=0. In this case, from (39), we have ${\mathop {{\left\| X \right\|}^{2} }\limits_{t \geqslant t_{0} } } &lt; \eta$, if

$$ \frac{{\varepsilon _{{\text{s}}} }} {{\zeta _{{\text{s}}} }} \leqslant \eta \lambda _{{\min }} (P) $$

(42)

In the case of nonzero initial states, V(t ₀)≠0. We can obtain ${\mathop {{\left\| X \right\|}^{2} }\limits_{t \geqslant T} } &lt; \eta$, if

$$ V(t_{0} ){\text{e}}^{{ - \zeta _{{\text{s}}} (t - t_{0} )}} + \frac{{\varepsilon _{{\text{s}}} }} {{\zeta _{{\text{s}}} }} \leqslant \eta \lambda _{{\min }} (P) $$

(43)

If $T \geqslant t_{0} + \frac{1} {{\zeta _{{\text{s}}} }}\ln (\frac{{2V(t_{0} )}} {{\eta \lambda _{{\min }} (P)}})$, we have

$$ V(t_{0} ){\text{e}}^{{ - \zeta _{{\text{s}}} (T - t_{0} )}} \leqslant \frac{1} {2}\eta \lambda _{{\min }} (P) $$

Now to ensure (43) holds, it is required that

$$ \frac{{\varepsilon _{{\text{s}}} }} {{\zeta _{{\text{s}}} }} \leqslant \frac{1} {2}\eta \lambda _{{\min }} (P) $$

(44)

Incorporating (42) and (44) yields

$$ f_{i} \geqslant f_{{{\text{c}}4}} ,\quad f_{{{\text{c}}4}} = {\left( {\frac{{10k_{{{\text{q}}1}} \lambda _{{\max }} (P)}} {{\eta b_{0} \pi _{{\min }} \lambda _{{\min }} (P)}}} \right)}^{2} $$

(45)

From the previous analysis, we see that it is required

$$ f_{i} \geqslant {\mathop {\max }\limits_{j = 1}^4 }f_{{{\text{c}}j}} ,{\text{ }}\mu \geqslant \gamma _{{4\max }} + 2\gamma _{{5\max }} $$

where f _ci , i=1,...,4 are bounded constants and γ _4max+2γ _5max are bounded by a constant, all independent of f _i . When the parameters f _i and μ are set such that the above conditions hold, then the robust transient property and robust tracking property can be achieved. So the conclusions of the theorem are proven. □