METHOD OF OPTIMIZATION OF DYNAMIC SYSTEMS

A. I. Rubag

A method is proposed for calculating optimal controls for nonlinear dynamic objects:

\[ \dot{x}=f(x,t)+\varphi(x,t,u),\quad x(t_0)=x_0, \tag{1} \]

where \(u(t)\) belongs to a prescribed closed domain \(U\) and satisfies the criterion

\[ J=\int_{t_0}^{t_1}\bigl(Q(x,t)+\psi(t,u)\bigr)\,dt = \min_{u(t)\in U}. \tag{2} \]

Here \(x(t)\) and \(u(t)\) are, respectively, \(n\)- and \(m\)-dimensional column vectors of phase coordinates and control actions; \(Q(x,t)\) characterizes the quality of the control process; \(\psi(t,u)\) represents the losses caused by the operation of the control signals.

Let the positive definite function \(V(x,t)\) satisfy the linear partial differential equation

\[ \frac{\partial V(x,t)}{\partial t} + \frac{\partial V(x,t)}{\partial x}\,f(x,t) = -Q(x,t),\quad V(x(t_1),t_1)=0. \tag{3} \]

Then a necessary and sufficient condition for a minimum of the functional (2) is

\[ \frac{\partial V(x,t)}{\partial x}\,\varphi(x,t,u)+\psi(t,u) = \min_{u(t)\in U}. \tag{4} \]

Thus, if
\[ \varphi(x,t,u)=B(x,t)u(t),\quad \psi(t,u)=\frac12 u' D^{-1}u, \]
where \(D^{-1}(t)\) is the matrix inverse to the diagonal matrix of positive coefficients \(D(t)\), \(T\) is the transposition operation, and restrictions of the type \(|u(t)|\le H(t)\) are imposed on \(u(t)\), then it follows from (4) that

\[ u_{\mathrm{e}}(t)= \begin{cases} -D(t)B^T\left(\dfrac{\partial V}{\partial x}\right)^T, & \text{if }\left|DB^T\left(\dfrac{\partial V}{\partial x}\right)^T\right|\le H,\\[1.2em] -H\,\operatorname{sign}B^T\left(\dfrac{\partial V}{\partial x}\right)^T, & \text{if }\left|DB^T\left(\dfrac{\partial V}{\partial x}\right)^T\right|>H. \end{cases} \tag{5} \]

\[ \operatorname{sign} z=1,\ \text{if } z>0,\qquad \operatorname{sign} z=-1,\ \text{if } z<0. \]

For the case when \(f(x,t)=A(t)x(t)\), \(Q=\frac12 x^{T}(t)C(t)x(t)\), where \(C(t)\) is a diagonal matrix of positive coefficients, the positive definite function [1] \(V(x,t)\) should be sought in the form \(V=x^{T}\Gamma x\). In accordance with equation (3), the matrix \(\Gamma(t)\) is the solution of the linear differential equation

\[ \dot{\Gamma}=-(\Gamma+\Gamma^{T})A-C,\qquad \Gamma(t_1)=0. \tag{6} \]

The results obtained can also be used as an effective method for synthesizing regulators in the presence of constraints on the controls, on the basis of the use of the criterion

\[ J+V(x(t_1),t_1)=\min_{u(t)\in U}. \tag{7} \]

Let us consider an example: the synthesis of a regulator for the temperature regime of an ideal-mixing reactor in which a reaction of the type \(A+B\rightleftarrows C+D\) proceeds.

Notation: \(x_1(t),x_2(t)\) are, respectively, the concentrations of substances \(A\) and \(B\) at the reactor outlet; \(x_{10}(t),x_{20}(t)\) are those at the inlet; \(v_0(t)\) is the inlet-flow rate; \(\theta(t)\) is the volume of the reaction zone. Then the reactor equations take the form

\[ \dot{x}_i(t)=-K(t)x_1(t)x_2(t)+\bigl(x_{i0}(t)-x_i(t)\bigr)\frac{v_0(t)}{\theta(t)},\quad i=1,2,\quad K(t)=K_0\exp\left\{-\frac{E}{RT(t)}\right\}, \tag{8} \]

where the temperature has threshold constraints

\[ b_1\le T(t)\le b_2. \tag{9} \]

We take the desired control action to be

\[ u(t)=K(t)-\frac{(x_{10}-x_{20})v_0}{2x_1x_2\theta}, \tag{10} \]

with

\[ h_1(t)\le u(t)\le h_2(t),\qquad h_i=K_0\exp\left\{-\frac{E}{Rb_i}\right\}-\frac{(x_{10}-x_{20})v_0}{2x_1x_2\theta},\quad i=1,2. \tag{11} \]

If necessary, one can always pass from \(u(t)\) to the temperature \(T(t)\). Then, in accordance with (5), the regulator operating algorithm takes the form

\[ u_{\mathrm e}(t)= \begin{cases} z(t), & \text{if } h_1(t)\le z(t)\le h_2(t),\\ h_1(t), & \text{if } z(t)\le h_1(t),\\ h_2(t), & \text{if } z(t)\ge h_2(t), \end{cases} \tag{12} \]

\[ z(t)=4d(t)y(t)\bigl(x_1(t)+x_2(t)\bigr),\qquad \dot{y}(t)=2y(t)v_0(t)\theta^{-1}(t)-C(t), \]

and \(u_{\mathrm e}(t)\) ensures the minimum of the functional

\[ J=\frac12\int_{t_0}^{t_1}\left[(x_1+x_2)^2c+u^2d^{-1}\right]dt +V(x_1(t_1)+x_2(t_1),t_1),\qquad V(x,t)=x^2y. \]

Literature

1. A. A. Krasovskii. Izv. AN SSSR. Tekhnicheskaya kibernetika, No. 3, 164–175 (1970).