Absolute stability of control systems with retarded argument

By Lyapunov's direct method we investigate the behavior of systems of automatic control with one nonlinearity and retarded argument. Sufficient criteria for the absolute stability of systems for an arbitrary magnitude of the retardation are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 245–249, February, 1990.     

Modal control of multi-input systems of neutral type with retarded argument

We suggest a special choice of a basis in the linear span of columns of the “controllability matrix” for multi-input systems of neutral type with retarded argument, which permits one to obtain an effective sufficient solvability condition for the modal control problem for such systems. The proof of this condition provides a constructive way for designing the desired regulator by the feedback principle on the basis of the solution of linear algebraic systems over the ring of bivariate polynomials.     

On the Cauchy problem for quasilinear hyperbolic systems with a retarded argument

Summary Quasilinear hyperbolic systems with a retarded argument in the Schauder canonic form are considered, with Cauchy data, and several independent variables. The proof of existence and uniqueness theorem is based on recent results due to L. Cesari and P. Bassanini for systems without a retarded argument.     

Transformation of equations with retarded argument to equations with the best argument

The problem of choosing the best argument in the Cauchy problem for a system of ordinary differential equations with retarded argument is studied from the viewpoint of the method of continuation of the solution with respect to a parameter. It is proved that the arc length counted along the integral curve of the problem is the best argument for the system of continuation equations to be well-posed in the best possible way. A transformation of the Cauchy problem to the best argument is obtained.     

Transformation of equations with retarded argument to equations with the best argument

The problem of choosing the best argument in the Cauchy problem for a system of ordinary differential equations with retarded argument is studied from the viewpoint of the method of continuation of the solution with respect to a parameter. It is proved that the arc length counted along the integral curve of the problem is the best argument for the system of continuation equations to be well-posed in the best possible way. A transformation of the Cauchy problem to the best argument is obtained. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 62–68, January, 1998.     

Cubic-spline identification of distributed-parameter systems

The use of doubly cubic splines is presented for the identification of a general second-order distributed-parameter system. The application of doubly cubic splines is shown to result in a system of linear algebraic equations that may be solved for the unknown process parameters in an on-line recursive manner. Due to the advantageous properties of splines, the approximation of partial derivatives is shown to incorporate temporal and spatial smoothing, an important feature when process data is subject to random disturbances. Coefficients of extraneous terms in the assumed model are also correctly identified. A comparison of the spline technique with other identification schemes in current use indicates that the spline application is consistently superior with regard to accuracy of estimation and rate of convergence.     

Control of stochastic distributed-parameter systems

A scheme is proposed for the feedback control of distributed-parameter systems with unknown boundary and volume disturbances and observation errors. The scheme consists of employing a nonlinear filter in the control loop such that the controller uses the optimal estimates of the state of the system. A theoretical comparison of feedback proportional control of a styrene polymerization reactor with and without filtering is presented. It is indicated how an approximate filter can be constructed, greatly reducing the amount of computing required.Notation a(t) l-vector noisy dynamic input to system - A(t, a) l-vector function - A frequency factor for first-order rate law (5.68×10⁶ sec^–1) - b distance to the centerline between two coil banks in the reactor (4.7 cm) - B k-vector function defining the control action - c(, ) concentration of styrene monomer, molel ^–1 - C _p heat capacity (0.43 cal · g^–1 · K^–1) - C _ij constants in approximate filter, Eqs. (49)–(52) - E activation energy (20330 cal · mole^–1) - expectation operator - f(t,...) n-vector function - g ₀,g ₁(t,...) n-vector functions - h(t, u) m-vector function relating observations to states - H(t) function defined in Eq. (36) - k dimensionality of control vectorv(x, t) - k _i constants in approximate filter, Eqs. (49)–(52) - K dimensionless proportional gain - l dimensionality of dynamic inputa(t) - m dimensionality of observation vectory(t) - n dimensionality of state vectoru(x, t) - P ^(vv)(x, s, t) n×n matrix governed by Eq. (9) - P ^(va)(x, t) n×l matrix governed by Eq. (10) - P ^(aa)(t) l×l matrix governed by Eq. (11) - q _i (t) diagonal elements ofm×m matrixQ(x, s, t) - Q(x, s, t) m×m weighting matrix - R universal gas constant (1.987 cal · mole^–1 · K^–1) - R(x, s, t) n×n weighting matrix - R _i (t) n×n weighting matrix - s dimensionless spatial variable - S(x, s, t) matrix defined in Eq. (11) - t dimensionless time variable - T(, ) temperature, K - u(x, t) n-dimensional state vector - u _c (t) wall temperature - u _d desired value ofu ₁(1,t) - u _c * reference control value ofu _c - u _c ^max maximum value ofu _c - u _c ^min minimum value of _c - v(x, t) k-dimensional control vector - W(t) l×l weighting matrix - x dimensionless spatial variable - y(t) m-dimensional observation vector - _i constants in approximate filter, Eqs. (49)–(52) - dimensionless parameter defined in Eq. (29) - H heat of reaction (17500 cal · mole^–1) - dimensionless activation energy, defined in Eq. (29) - (x) Dirac delta function - (x, t) m-dimensional observation noise - thermal conductivity (0.43×10^–3 cal · cm^–1 · sec^–1 · K^–1) - density (1 g · cm^–3) - time, sec - dimensionless parameter defined in Eq. (29) - spatial variable, cm - * reference value - ^ estimated value     

Stability estimates for linear stochastic systems with deviating argument of neutral type

We study linear stochastic differential equations with deviating argument of neutral type and establish sufficient conditions of stability. The functions determining the initial perturbations of solutions are found.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 834–842, June, 1993.     

Control of non-self-adjoint distributed-parameter systems

Systems involving viscous damping forces, circulatory forces, and aerodynamic forces are non-self-adjoint. A method capable of controlling non-self-adjoint distributed systems is the independent modal-space control method, whereby the problem of controlling a distributed-parameter system is reduced to that of controlling an infinite set of independent, complex, second-order ordinary differential equations. In the case of optimal control, one must solve independent, complex, scalar Riccati equations. The transient solution of the Riccati equations can be found with relative ease and the steady-state solution can be found in closed form. A numerical example demonstrates the effectiveness of the method.This work was supported by AFOSR Research Grant No. 83-0017.