Suboptimal Robust Stabilization of an Unknown Autoregressive Object with Uncertainty and Offset External Perturbation

Abstract

In this paper, the problem of suboptimal stabilization of an object with discrete time, output and control uncertainties, and bounded external perturbation is considered. The autoregressive nominal model coefficients, uncertainty amplification coefficients, norm and external disturbance offset are assumed to be unknown. The quality indicator is the worst-case asymptotic upper bound of the output modulus of the object. The solution of the problem in conditions of non-identifiability of all unknown parameters is based on the method of recurrent target inequalities and optimal online estimation, in which the quality index of the control problem serves as an identification criterion. A non-linear replacement of the unknown parameter perturbations that reduces the optimal online estimation problem to a fractional linear programming problem is proposed. The performance of adaptive suboptimal control is illustrated by numerical simulation results.

APPENDIX

Proof of Theorem 1. The robust stability condition (3.3) follows from Theorem 7 [24] applied to system (2.1), (3.1). To prove the second statement of the theorem, it is sufficient to apply Theorems 5 and 6 [24] (see also [15]). To do this, we have to present the system (2.1), (3.1) in the standard M–Δ form given in Fig. 5 and having a block form

$$\left( \begin{gathered} y \\ z \\ \end{gathered} \right) = M\left( \begin{gathered} r \\ w \\ \xi \\ \end{gathered} \right) = \left( {\begin{array}{*{20}{c}} {{{M}_{{yr}}}}&{{{M}_{{yw}}}}&{{{M}_{{y\xi }}}} \\ {{{M}_{{zr}}}}&{{{M}_{{zw}}}}&{{{M}_{{z\xi }}}} \end{array}} \right)\left( \begin{gathered} r \\ w \\ \xi \\ \end{gathered} \right),\quad \xi = \Delta z.$$

(A.1)

Fig. 5.

M–Δ form of the system (2.1), (3.1).

For the system (2.1), (3.1) the signal r = c^w1, 1 = (1, 1, …), and M–Δ form looks like

$$\left( \begin{gathered} y \\ {{z}^{1}} \\ {{z}^{2}} \\ \end{gathered} \right) = M\left( \begin{gathered} {\mathbf{1}} \\ w \\ {{\xi }^{1}} \\ {{\xi }^{2}} \\ \end{gathered} \right) = \left( {\begin{array}{*{20}{c}} 0&{{{\delta }^{w}}}&{{{\delta }^{y}}}&{{{\delta }^{u}}} \\ 0&{{{\delta }^{w}}}&{{{\delta }^{y}}}&{{{\delta }^{u}}} \\ { - \frac{{{{c}^{w}}q}}{{{{b}_{1}}}}}&{{{\delta }^{w}}{{G}^{\xi }}({{q}^{{ - 1}}})}&{{{\delta }^{y}}{{G}^{\xi }}({{q}^{{ - 1}}})}&{{{\delta }^{u}}{{G}^{\xi }}({{q}^{{ - 1}}})} \end{array}} \right)\left( \begin{gathered} {\mathbf{1}} \\ w \\ {{\xi }^{1}} \\ {{\xi }^{2}} \\ \end{gathered} \right),$$

(A.2)

where

$${{z}_{t}} = \left( \begin{gathered} {{y}_{t}} \\ {{u}_{t}} \\ \end{gathered} \right),\quad \xi = \left( \begin{gathered} {{\xi }^{1}} \\ {{\xi }^{2}} \\ \end{gathered} \right) = \left( {\begin{array}{*{20}{c}} {{{\Delta }^{1}}}&0 \\ 0&{{{\Delta }^{2}}} \end{array}} \right)z = \left( \begin{gathered} {{\Delta }^{1}}(y) \\ {{\Delta }^{2}}(u) \\ \end{gathered} \right).$$

The first and second lines of the matrix M in (A.2) correspond to Eq. (3.2). The third row of M corresponds to the representation of the optimal regulator (3.1) in the form of

$${{u}_{t}} = - {{c}^{w}}{\text{/}}{{b}_{1}} + {{G}^{\xi }}({{q}^{{ - 1}}}){{y}_{t}} = - {{c}^{w}}{\text{/}}{{b}_{1}} + {{\delta }^{w}}{{G}^{\xi }}({{q}^{{ - 1}}}){{w}_{t}} + {{\delta }^{y}}{{G}^{\xi }}({{q}^{{ - 1}}})\xi _{t}^{1} + {{\delta }^{u}}{{G}^{\xi }}({{q}^{{ - 1}}})\xi _{t}^{2}.$$

The formula for J(θ) in (3.4) corresponds to the quality index (2.4), in which sup is taken on the perturbation set ${v}$ with uncertainties Δ¹ and Δ² with finite memory (see [12]), and is derived by Theorem 5 [24] as follows. Let us assume that ||z||_ss = (||z¹||_ss, …, ||z ^p||_ss)^T for the vector sequence z ∈ $\ell _{e}^{p}$, and

$${{[M]}_{1}}: = \left( {\begin{array}{*{20}{c}} {{{{\left\| {{{M}_{{11}}}} \right\|}}_{1}}}& \ldots &{{{{\left\| {{{M}_{{1p}}}} \right\|}}_{1}}} \\ \vdots & \vdots & \vdots \\ {{{{\left\| {{{M}_{{q1}}}} \right\|}}_{1}}}& \ldots &{{{{\left\| {{{M}_{{qp}}}} \right\|}}_{1}}} \end{array}} \right)$$

for a stable q × p response matrix M of impulses M_ij ∈ ${{\ell }_{1}}$. For the matrix M from (A.1) we will assume that

$${{M}_{{ss}}}(r): = \left( {\begin{array}{*{20}{c}} {{{{[{{M}_{{yr}}}r]}}_{{ss}}} + {{{[{{M}_{{yw}}}]}}_{1}}}&{{{{[{{M}_{{y\xi }}}]}}_{1}}} \\ {{{{[{{M}_{{zr}}}r]}}_{{ss}}} + {{{[{{M}_{{zw}}}]}}_{1}}}&{{{{[{{M}_{{z\xi }}}]}}_{1}}} \end{array}} \right).$$

According to Theorem 5 from [24]

$$J(\theta ) = {{[{{M}_{{yr}}}r]}_{{ss}}} + {{[{{M}_{{yw}}}]}_{1}} + {{[{{M}_{{y\xi }}}]}_{1}}{{(I - {{[{{M}_{{z\xi }}}]}_{1}})}^{{ - 1}}}({{[{{M}_{{zr}}}r]}_{{ss}}} + {{[{{M}_{{zw}}}]}_{1}}).$$

Then, for the system (A.2), we have

$$J(\theta ) = {{\delta }^{w}} + \left( {{{\delta }^{y}}\;{{\delta }^{u}}} \right){{\left( {I - \left( {\begin{array}{*{20}{c}} {{{\delta }^{y}}}&{{{\delta }^{u}}} \\ {{{\delta }^{y}}\left\| {{{G}^{\xi }}} \right\|}&{{{\delta }^{u}}\left\| {{{G}^{\xi }}} \right\|} \end{array}} \right)} \right)}^{{ - 1}}}\left( \begin{gathered} {{\delta }^{w}} \\ \left| {{{c}^{w}}} \right|{{\left\| {\frac{{ - q}}{{{{b}_{1}}}}r} \right\|}_{{ss}}} + {{\delta }^{w}}\left\| {{{G}^{\xi }}} \right\| \\ \end{gathered} \right)$$

$$ = {{\delta }^{w}} + \left( {{{\delta }^{y}}\;{{\delta }^{u}}} \right){{\left( {\begin{array}{*{20}{c}} {1 - {{\delta }^{y}}}&{ - {{\delta }^{u}}} \\ { - {{\delta }^{y}}\left\| {{{G}^{\xi }}} \right\|}&{1 - {{\delta }^{u}}\left\| {{{G}^{\xi }}} \right\|} \end{array}} \right)}^{{ - 1}}}\left( \begin{gathered} {{\delta }^{w}} \\ \left| {{{c}^{w}}{\text{/}}{{b}_{1}}} \right| + {{\delta }^{w}}\left\| {{{G}^{\xi }}} \right\| \\ \end{gathered} \right)$$

$$ = {{\delta }^{w}} + \frac{1}{{1 - {{\delta }^{y}} - {{\delta }^{u}}\left\| {{{G}^{\xi }}} \right\|}}\left( {{{\delta }^{y}}\;{{\delta }^{u}}} \right)\left( {\begin{array}{*{20}{c}} {1 - {{\delta }^{u}}\left\| {{{G}^{\xi }}} \right\|}&{{{\delta }^{u}}} \\ {{{\delta }^{y}}\left\| {{{G}^{\xi }}} \right\|}&{1 - {{\delta }^{y}}} \end{array}} \right)\,{\kern 1pt} \left( \begin{gathered} {{\delta }^{w}} \\ \left| {{{c}^{w}}{\text{/}}{{b}_{1}}} \right| + {{\delta }^{w}}\left\| {{{G}^{\xi }}} \right\| \\ \end{gathered} \right)$$

$$ = {{\delta }^{w}} + \frac{1}{{1 - {{\delta }^{y}} - {{\delta }^{u}}\left\| {{{G}^{\xi }}} \right\|}}\left( {{{\delta }^{y}}\;{{\delta }^{u}}} \right)\left( \begin{gathered} {{\delta }^{w}} + {{\delta }^{u}}\left| {{{c}^{w}}{\text{/}}{{b}_{1}}} \right| \\ (1 - {{\delta }^{y}})\left| {{{c}^{w}}{\text{/}}{{b}_{1}}} \right| + {{\delta }^{w}}\left\| {{{G}^{\xi }}} \right\| \\ \end{gathered} \right)$$

$$ = {{\delta }^{w}} + \frac{{{{\delta }^{y}}{{\delta }^{w}} + {{\delta }^{u}}\left| {{{c}^{w}}} \right|\left\| {{{G}^{\xi }}} \right\| + {{\delta }^{u}}{{\delta }^{w}}\left\| {{{G}^{\xi }}} \right\|}}{{1 - {{\delta }^{y}} - {{\delta }^{u}}\left\| {{{G}^{\xi }}} \right\|}} = \frac{{{{\delta }^{w}} + {{\delta }^{u}}\left. {\left| {{{c}^{w}}{\text{/}}{{b}_{1}}} \right|} \right)}}{{1 - {{\delta }^{y}} - {{\delta }^{u}}\left\| {{{G}^{\xi }}} \right\|}}.$$

Finally, the monotonic convergence of J_μ(θ) to J(θ) in (3.4) is guaranteed by Theorem 6 from [24].

Proof of Statement 1. The vector $\hat {\theta }$ satisfies the a priori assumption AP1 due to the conditions of Statement 1. For all t > 0, let us assume ${{{\hat {v}}}_{t}}$ = $\hat {a}({{q}^{{ - 1}}}){{y}_{t}}$ – ${{\hat {b}}_{1}}{{u}_{{t - 1}}}$. Then the control object with the parameter vector $\hat {\theta }$ and the total perturbation ${\hat {v}}$ satisfies Eq. (2.1), and due to (4.1) the perturbation ${\hat {v}}$ satisfies the inequalities

$$\left| {{{{{\hat {v}}}}_{t}} - {{{\hat {c}}}^{w}}} \right|\;\leqslant \;{{\hat {\delta }}^{w}} + {{\hat {\delta }}^{y}}p_{t}^{y} + {{\hat {\delta }}^{u}}p_{t}^{u}.$$

The values of ${{{\hat {v}}}_{t}}$ can be represented in the form of (2.2) by choosing suitable values of w_t, Δ¹(y)_t, Δ²(u)_t that satisfy the inequalities (2.3), and thereby ensure that the a priori assumption of AP2 is true.

Proof of Theorem 2. Let us prove that for each update of the estimates, the distance from ζ_t to the half-space Ω_t+1 is greater than ε. Since ζ_t only changes when $\psi _{{{\text{t + 1}}}}^{{\text{T}}}{{\zeta }_{t}}$ < ${{\nu }_{{t + 1}}}$ – $\varepsilon \left| {{{\psi }_{{t + 1}}}} \right|$ and $\psi _{{t + 1}}^{{\text{T}}}\hat {\zeta }$ $\geqslant$ ${{\nu }_{{t + 1}}}$ for all $\hat {\zeta } \in {{\Omega }_{{t + 1}}}$, then

$$\varepsilon \left| {{{\psi }_{{t + 1}}}} \right| < \left| {\psi _{{t + 1}}^{{\text{T}}}(\hat {\zeta } - {{\zeta }_{t}})} \right|\;\leqslant \;\left| {{{\psi }_{{t + 1}}}} \right|\left| {\hat {\zeta } - {{\zeta }_{t}}} \right|$$

and, therefore, $\left| {\hat {\zeta } - {{\zeta }_{t}}} \right| > \varepsilon$ for all $\hat {\zeta } \in {{\Omega }_{{t + 1}}}$. Thus, after adding the inequality $\psi _{{t{\text{ + 1}}}}^{{\text{T}}}\hat {\zeta }$ $\geqslant$ ${{\nu }_{{t + 1}}}$ describing the half-space Ω_t+1 to the description of Z_t, the polyhedron Z_t+1 and all subsequent ones do not intersect the neighborhood of ε of the vector ζ_t ∈ Z_t. It follows that the ε/2-neighborhoods of the various estimates of ζ_t do not intersect each other. Since ${{Z}_{{t + 1}}} \subset {{Z}_{t}}$ for all t, the number of changes in the estimates of Z_t and ζ_t will be finite if the estimates of ζ_t lie in a bounded set. From the equation of the adaptive regulator (6.1) for all t we have

$$\left| {{{u}_{t}}} \right|\;\leqslant \;\left| {c_{t}^{w}{\text{/}}b_{1}^{t}} \right| + \left\| {{{G}^{{{{\xi }_{t}}}}}} \right\|\left| {y_{{t - n + 1}}^{t}} \right|.$$

Then, for the object (2.1) on the time interval [0, t], the inequalities (5.5) with the parameters

$$\tilde {\delta }_{t}^{e} = {{\delta }^{w}} + {{\delta }^{u}}\mathop {\max }\limits_{s\;\leqslant \;t} \left| {c_{s}^{w}{\text{/}}b_{1}^{s}} \right|,\quad {{\tilde {\delta }}_{t}} = {{\delta }^{y}} + {{\delta }^{u}}\mathop {\max }\limits_{s\;\leqslant \;t} \left\| {{{G}^{{{{\xi }_{s}}}}}} \right\|$$

are true. Therefore, ${{\tilde {\zeta }}_{t}}$ = (ξ^T, c^w, $\tilde {\delta }_{t}^{e}$, ${{\tilde {\delta }}_{t}}$)^T ∈ Z_t for all t. If the assumption (6.7) is satisfied, then I(${{\tilde {\zeta }}_{t}}$) $\leqslant$ $\bar {I}$, where $\bar {I}$ is defined in (6.9) (with the right-hand inequality in (6.9) obviously followed from the definition of G_u in (6.6)). From (6.5) for all t, it follows that

$$I({{\zeta }_{t}})\;\leqslant \;I({{\tilde {\zeta }}_{t}})$$

and then I(ζ_t) $\leqslant$ $\bar {I}$. From the boundedness of I(ζ_t), there follows the boundedness of the estimates ζ_t and thus the finiteness of the number of updates of the estimates ξ_t  and Z_t. Then ζ_t = ζ_∞ = ($\xi _{\infty }^{{\text{T}}}$, $c_{\infty }^{w}$, $\delta _{\infty }^{e}$, ${{\delta }_{\infty }}$) from some point of time t_∞ and

$$\psi _{{t + 1}}^{{\text{T}}}{{\zeta }_{\infty }}\; \geqslant \;{{\nu }_{{t + 1}}} - \varepsilon \left| {{{\psi }_{{t + 1}}}} \right|\quad \forall t\; \geqslant \;{{t}_{\infty }}.$$

(A.3)

From (A.3), it follows that, for all t $\geqslant$ t_∞,

$$\left| {{{a}_{\infty }}({{q}^{{ - 1}}}){{y}_{{t + 1}}} - {{b}_{\infty }}({{q}^{{ - 1}}}){{u}_{t}} - c_{\infty }^{w}} \right|\;\leqslant \;\delta _{\infty }^{e} + {{\delta }_{\infty }}{{p}_{{t + 1}}} + \varepsilon \left| {{{\psi }_{{t + 1}}}} \right|$$

$$\leqslant \;\delta _{\infty }^{e} + {{\delta }_{\infty }}{{p}_{{t + 1}}} + \varepsilon \left( {\sqrt {n + 1} {{p}_{{t + 1}}} + \sqrt 2 + \left| {{{u}_{t}}} \right|} \right)$$

$$\leqslant \;\delta _{\infty }^{e} + \varepsilon \left( {\sqrt 2 + \left| {c_{\infty }^{w}{\text{/}}b_{1}^{\infty }} \right|} \right) + \left[ {{{\delta }_{\infty }} + \varepsilon \left( {\sqrt {n + 1} + \left\| {{{G}^{{{{\xi }_{\infty }}}}}} \right\|} \right)} \right]{{p}_{{t + 1}}}.$$

Given Statement 1 of Section 4, it follows from the obtained inequality that the output of y at all t $\geqslant$ ${{t}_{*}}$ satisfies the Eq. (2.1) with the parameter vector $\zeta _{\infty }^{\varepsilon }$ of the form (6.10). Then Theorem 1 guarantees the left-hand inequality in (6.8). To prove the right-hand inequality in (6.8), we estimate the difference I($\zeta _{\infty }^{\varepsilon }$) – I(${{\zeta }_{\infty }}$) from above using the inequality

$$\frac{{{{C}_{1}} + {{\varepsilon }_{1}}}}{{{{C}_{2}} - {{\varepsilon }_{2}}}} - \frac{{{{C}_{1}}}}{{{{C}_{2}}}} = \frac{{{{C}_{2}}{{\varepsilon }_{1}} + {{C}_{1}}{{\varepsilon }_{2}}}}{{{{C}_{2}}({{C}_{2}} - {{\varepsilon }_{2}})}} < \frac{{{{\varepsilon }_{1}} + {{C}_{1}}{{\varepsilon }_{2}}}}{{{{{({{C}_{2}} - {{\varepsilon }_{2}})}}^{2}}}}$$

with the parameters C₁ = $\delta _{\infty }^{e}$, C₂ = 1 – δ_∞ $\leqslant$ 1, ε₁ = ε($\sqrt 2$ + |$c_{\infty }^{w}{\text{/}}b_{1}^{\infty }$|), ε₂ = ε($\sqrt {n + 1}$ + ||${{G}^{{{{\xi }_{\infty }}}}}$||). Then

$$I(\zeta _{\infty }^{\varepsilon }) - I({{\zeta }_{\infty }}) < \frac{{\sqrt 2 + \left| {c_{\infty }^{w}{\text{/}}b_{1}^{\infty }} \right| + \delta _{\infty }^{e}\left( {\sqrt {n + 1} + \left\| {{{G}^{{{{\xi }_{\infty }}}}}} \right\|} \right)}}{{{{{\left( {1 - {{\delta }_{\infty }} - \varepsilon \left( {\sqrt {n + 1} + \left\| {{{G}^{{{{\xi }_{\infty }}}}}} \right\|} \right)} \right)}}^{2}}}}\varepsilon $$

and, therefore, ${{K}_{{{{\zeta }_{\infty }}}}}$ has the form of (6.11). The first statement of Theorem 2 is proved.

Let us prove the second statement. Now let the inequalities (6.12) be satisfied in the closed adaptive system. Then inequalities (5.3) with constants C₁ = |c^w/b₁| and C₂ = ||G ^ξ|| follow from the object Eq. (2.1). This means that for the unknown parameter vector ζ defined in (5.9), the target inequalities (5.5) with the parameters δ^e, δ of the form (5.8) and inclusion ζ ∈ Z_t are satisfied for all t. Then, due to the choice of optimal estimates ζ_t according to (6.5), at all t,

$$I({{\zeta }_{t}})\;\leqslant \;I(\zeta ) = J(\theta ),$$

where the equality I(ζ) = J(θ) is established in (5.9). Hence, as in the first statement of Theorem 2, there follows the convergence of the estimates ξ_t and Z_t in finite time and the inequalities (6.13).