Parametric Optimization of Nonlinear Systems Represented by Models Using the Extended Linearization Method

Abstract

We state an optimal control problem for a class of dynamical systems whose nonlinear objects can be represented as objects with linear structure and state-dependent parameters. The linearity of the structure of the transformed nonlinear system and the quadratic performance functional allow one to move from the need to search for solutions of the Hamilton–Jacobi equation to an equation of the Riccati type with state-dependent parameters when synthesizing the optimal control, i.e., the controller parameters. The main problem of implementing the optimal control is related to the problem of being capable of finding solutions of such an equation online, at the object operation rate. An algorithmic method for the parametric optimization of the controller is proposed. The method is based on using the necessary optimality conditions for the control system in question. Our algorithms can be used both to optimize the time-varying objects themselves given an appropriate choice of the parameters for this purpose and to optimize the entire control system using an appropriate parametric adjustment of the controllers. The efficiency of the algorithms is demonstrated by the example of drug treatment of patients with HIV.

APPENDIX

Proof of Theorem 1. Let us substitute the expression $d\left [x^{\mathrm {T}} (t) S(x(t))x(t)\right ]/dt$ into the integrand of the functional

$$ J\big (x(\cdot ),u(\cdot )\big )=\frac {1}{2} x^{\mathrm {T}} (t_{f} ) Fx(t_{f} )+\frac {1}{2} \int \limits _{t_{0} }^{t_{f} }\big \{x^{\mathrm {\; T}} (t) Qx(t)+u^{\mathrm {\; T}} (t)R u(t)\big \} dt,$$

having compensating this outside the integral by the relation

$$ 0.5\Big [x^{\mathrm {T}}(t)S\big (x(t)\big )x(t)-x^{\mathrm {T}}(t_{f}) S\big (x(t_{f})\big )x(t_{f})\Big ]. $$

We obtain

$$ \begin {aligned} J\big (x(\cdot ),u(\cdot )\big )&=\frac {1}{2} x^{\mathrm {T}} (t_{f} ) Fx(t_{f} )+\frac {1}{2} \Big [x^{\mathrm {T}} (t) S\big (x(t)\big )x(t)-x^{\mathrm {T}} (t_{f} ) S(x(t_{f} ))x(t_{f} )\Big ] \\ &\qquad {}+\frac {1}{2} \int \limits _{t}^{t_{f} }\bigg \{x^{\mathrm {\; T}} (t) Qx(t)+u^{\mathrm {\; T}} (t)R u(t)+ d\Big [x^{\mathrm {T}} (t) S\big (x(t)\big )x(t)\Big ]/dt\bigg \} dt. \end {aligned}$$

Since

$$\frac {d}{dt} x(t)=\Big [\big (x(t)\big )-g\big (x(t)\big )R^{-1} g^{\mathrm {T}} \big (x(t)\big )S\big (x(t)\big )\Big ]x(t), \quad x(t_{0} )=x_{0},$$

(A.1)

and $S(x(t_{f} ))=F$, we have

$$ J^{0 } \big (t,x(t)\big )= \frac {1}{2} x^{\mathrm {T}} (t)S\big (x(t)\big )x(t), \quad t_{0} \le t\le t_{f} .\quad \blacksquare$$

Proof of 2. To prove the theorem on the asymptotic stability of the model (A.1), we introduce a Lyapunov function $V_{L} (x(t))$ such that

$$ \omega _{1} \big \{|x|\big \}\le V_{L} \big (x(t)\big )\le \omega _{2} \big \{|x|\big \},\quad dV_{L} \big (x(t)\big )/dt\le -\omega _{3} \big \{|x|\big \} \quad \forall x,$$

(A.2)

where $\omega _{i} \left \{\left |x\right |\right \}$, $i=1,2,3$, are scalar nondecreasing functions such that $\omega _{i} (0)=0$, $\omega _{i} \left \{\left |x\right |\right \}>0$. Using the second Lyapunov theorem, we conclude that the system is stable under the condition

$$ \frac {dV_{L} \big (x(t)\big )}{dt} =\frac {\partial V_{L} \big (x(t)\big )}{\partial x} \frac {dx(t)}{dt} \le -\omega _{3} \big \{|x|\big \}.$$

(A.3)

Define $V_{L} (x(t))$ in the form

$$ {V_{L} \big (x(t)\big )=x^{\mathrm {T}} (t)S\big (x(t)\big )x(t)},$$

where $S(x(t))$ is a positive definite symmetric matrix that is a solution of the Riccati equation with state-dependent parameters,

$$ \begin {gathered} \eqalign { \frac {dS\big (x(t)\big )}{dt} &+S\big (x(t)\big )A\big (x(t)\big )+A^{\mathrm {T}} \big (x(t)\big )S\big (x(t)\big )\cr &{}-S\big (x(t)\big )g\big (x(t)\big )R^{-1} g^{\mathrm {T}} \big (x(t)\big )S\big (x(t)\big )+Q=0,} \\ {S\big (x(t_{f})\big )=F.} \end {gathered} $$

(A.4)

Define $\omega _{3} \left \{\left |x\right |\right \}$ in the form $\omega _{3} \left \{\left |x\right |\right \}=x^{\mathrm {T}} (t)Qx(t)$, $\forall x\ne 0$. Then, in view of (A.1), condition (A.3) must be satisfied,

$$ \begin {aligned} \frac {dV_{L} \big (x(t)\big )}{dt}&=x^{\mathrm {T}} (t)\bigg [\frac {dS\big (x(t)\big )}{dt} +S\big (x(t)\big )A\big (x(t)\big )+A^{\mathrm {T}} \big (x(t)\big )S\big (x(t)\big ) \\ &\qquad \qquad {}-S\big (x(t)\big )g\big (x(t)\big )R^{\mathrm {-1}} g^{\mathrm {T}} \big (x(t)\big )S\big (x(t)\big ) +Q\bigg ]x(t)\\ &\qquad \qquad \qquad {}-x^{\mathrm {T}} (t)S\big (x(t)\big )g\big (x(t)\big )R^{\mathrm {-1}} g^{\mathrm {T}} \big (x(t)\big )S\big (x(t)\big )x(t)\le 0. \end {aligned} $$

Taking into account (A.4), we have

$$ x^{\mathrm {T}} (t)S\big (x(t)\big )g\big (x(t)\big )R^{\mathrm {-1}} g^{\mathrm {T}} \big (x(t)\big )S\big (x(t)\big )x(t)\ge 0. $$

This condition is satisfied for all $x(t)\ne 0$. Consequently, the model (A.1) of the nonlinear system (2.1) is asymptotically stable. $\quad \blacksquare$

Proof of Theorem 5. To construct a parametric optimization algorithm for the system

$$ \frac {d}{dt} x(t)=\widetilde {f}\big (x(t),u(t),\eta (t),a(t)\big ),\quad x(t_{0} )=x_{0},$$

(A.5)

we introduce the Lyapunov function

$$ \eqalign { &V_{L} (\eta (t),a(t))\cr &\qquad {}=\frac {1}{2} \left \{\mathfrak {R} \left (x(t),u(t),\frac {\partial V\big (x(t)\big )}{\partial x} ,\eta (t),a(t)\right )-\mathfrak {R} \left (x^{0} (t),u^{0} (t),\frac {\partial V\big (x(t)\big )}{\partial x}\right ) \right \}^{2} \cr &\qquad {}=\frac {1}{2} \mathfrak {R} ^{2} \left (x(t),u(t),\frac {\partial V\big (x(t)\big )}{\partial x} ,\eta (t),a(t)\right ).}$$

(A.6)

Then for the asymptotic parametric optimization the derivative of this function must be negative for the case of $a(t)-\eta (t)\ne 0$,

$$ \begin {aligned} \frac {d}{dt} V_{L} (\eta (t),a(t))&=\mathfrak {R} \left (x(t),u(t),\frac {\partial V\big (x(t)\big )}{\partial x} ,\eta (t),a(t)\right )\\ &\qquad {}\times \left [\frac {\partial H\left (x(t),u(t),\frac {\partial V\big (x(t)\big )}{\partial x} ,\eta (t),a(t)\right )}{\partial \eta } \frac {d}{dt} \eta (t)\right .\\ &\qquad \qquad {}+\left .{}\frac {\partial H\left (x(t),u(t),\frac {\partial V\big (x(t)\big )}{\partial x} ,\eta (t),a(t)\right )}{\partial a} \frac {d}{dt} a(t) \right ]<0, \end {aligned}$$

(A.7)

because $\displaystyle \frac {\partial H}{\partial t} =\frac {\partial H^{0} }{\partial t} =0$, $\displaystyle \frac {\partial \varphi (t)}{\partial \eta } =0$ and $\displaystyle \frac {\partial \varphi (t)}{\partial a} =0$.

Define a parametric optimization algorithm in the form

$$ \begin {gathered} \frac {d}{dt} a(t)=-\left \{\frac {\partial H\left (x(t),u(t),\frac {\partial V\big (x(t)\big )}{\partial x} ,\eta (t),a(t)\right )}{\partial a} \right \}^{\mathrm {T}}\mathfrak {R} \left (x(t),u(t),\frac {\partial V\big (x(t)\big )}{\partial x} ,\eta (t),a(t)\right ), \\ a(t_{0} )=a_{0} . \end {gathered} $$

With this assignment of the parametric optimization algorithm, from condition (A.7) we obtain

$$ \eqalign { &\mathfrak {R} \left (x(t),u(t),\frac {\partial V\big (x(t)\big )}{\partial x} ,\eta (t),a(t)\right )\frac {\partial H\left (x(t),u(t),{\partial V\big (x(t)\big )}\big /{\partial x} ,\eta (t),a(t)\right )}{\partial \eta } \frac {d}{dt} \eta (t)\cr &\qquad {}-\mathfrak {R} ^{2} \left (x(t),u(t),\frac {\partial V\big (x(t)\big )}{\partial x} ,\eta (t),a(t)\right )\frac {\partial H\left (x(t),u(t),{\partial V\big (x(t)\big )}\big /{\partial x} ,\eta (t),a(t)\right )}{\partial a} \cr &\qquad \qquad {}\times \left \{\frac {\partial H\left (x(t),u(t),{\partial V\big (x(t)\big )}\big /{\partial x} ,\eta (t),a(t)\right )}{\partial a} \right \}<0.} $$

Since the rate of change of disturbances is subject to a constraint, i.e.,

$$ {\left |d\eta (t)\big / dt \right |\le \max \nolimits _{\eta \in \Delta } \left |d\eta (t)\big / dt\right |=\sigma },\quad {\sigma _{i}>0},\quad {i=1,\ldots ,k}, $$

we can write the condition for the success of the optimization process as

$$ \eqalign { &\mathfrak {R}^{2} \left (x(t),u(t),\frac {\partial V\big (x(t)\big )}{\partial x} ,\eta (t),a(t)\right )\left \| \frac {\partial H\left (x(t),u(t),{\partial V\big (x(t)\big )}\big /{\partial x} ,\eta (t),a(t)\right )}{\partial a} \right \| ^{2}\cr &\qquad {}>\left |\,\mathfrak {R} \left (x(t),u(t),\frac {\partial V\big (x(t)\big )}{\partial x} ,\eta (t),a(t)\right )\frac {\partial H\left (x(t),u(t),{\partial V\big (x(t)\big )}\big /{\partial x} ,\eta (t),a(t)\right )}{\partial \eta } \sigma \, \right |.\quad \blacksquare }$$