Stabilization of distributed control systems with delay

In this paper the asymptotic stabilization of linear distributed parameter control systems with delay is considered. Specifically, we are concerned with the class of control systems described by the equation x^′(t)=Ax(t)+L(x_t)+Bu(t), where A is the infinitesimal generator of a strongly continuous semigroup on a Banach space X. Assuming appropriate conditions, we will show that the usual spectral controllability assumption implies the feedback stabilization of the system. Applications to systems described by partial differential equations with delay are given.     

The formula of the solution for some classes of initial boundary value problems for the hyperbolic equation with two independent variables

A new representation is proved of the solutions of initial boundary value problems for the equation of the form u _xx (x, t) + r(x)u _x (x, t) ? q(x)u(x, t) = u _tt (x, t) + μ(x)u _t (x, t) in the section (under boundary conditions of the 1st, 2nd, or 3rd type in any combination). This representation has the form of the Riemann integral dependent on the x and t over the given section.     

Exact solution of mixed problems for variable coefficient one-dimensional diffusion equation

This paper deals with diffusion problems modeled by the equation a(t)u_xx = u_t, x > 0, t > 0, u(x, 0) = c(x) together with the boundary condition u(0, t) = b(t) or u_x(0, t) = b(t). By using Fourier transforms, existence conditions and exact solutions of the above mixed problems are given.     

A new critical theorem for adaptive nonlinear stabilization

It is fairly well known that there are fundamental differences between adaptive control of continuous-time and discrete-time nonlinear systems. In fact, even for the seemingly simple single-input single-output control system y_t+1=θ₁f(y_t)+u_t+w_t+1 with a scalar unknown parameter θ₁ and noise disturbance {w_t}, and with a known function f(⋅) having possible nonlinear growth rate characterized by |f(x)|=Θ(|x|^b) with b≥1, the necessary and sufficient condition for the system to be globally stabilizable by adaptive feedback is b<4. This was first found and proved by Guo (1997) for the Gaussian white noise case, and then proved by Li and Xie (2006) for the bounded noise case. Subsequently, a number of other types of “critical values” and “impossibility theorems” on the maximum capability of adaptive feedback were also found, mainly for systems with known control parameter as in the above model. In this paper, we will study the above basic model again but with additional unknown control parameter θ₂, i.e., u_t is replaced by θ₂u_t in the above model. Interestingly, it turns out that the system is globally stabilizable if and only ifb<3. This is a new critical theorem for adaptive nonlinear stabilization, which has meaningful implications for the control of more general uncertain nonlinear systems.     

Algorithmic complexity bounds on future prediction errors

We bound the future loss when predicting any (computably) stochastic sequence online. Solomonoff finitely bounded the total deviation of his universal predictor M from the true distribution μ by the algorithmic complexity of μ. Here we assume that we are at a time t > 1 and have already observed x = x₁ ⋯ x_t. We bound the future prediction performance on x_t+1x_t+2 ⋯ by a new variant of algorithmic complexity of μ given x, plus the complexity of the randomness deficiency of x. The new complexity is monotone in its condition in the sense that this complexity can only decrease if the condition is prolonged. We also briefly discuss potential generalizations to Bayesian model classes and to classification problems.     

Optimal controls for stochastic systems with singular noise

Stochastic control systems of the form dx₁ = f(t, x, u_t)dt + g(t, x)db_t (0 t 1), with g singular and general cost, are discussed. It is shown that there is an optimal relaxed control u that depends on the past of x and the driving process b. Nonstandard methods are used.     

Approximate Solutions of Polynomial Equations

In this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x)  ∈ Q [ x ], with its constant term equal to 0, such that F(x, y)  = ∑_j = 0ⁿc_j(y − g(x))^jfor some rational numbers c_j, hence, F(x, g(x)  + a)  ∈ Q for all a ∈ Q, or there are at most t distinct polynomials g₁(x),⋯ , g_t(x), t ≤ n, such that F(x, g_i(x))  ∈ Q for 1  ≤ i ≤ t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g₁(x),⋯ , g_t(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).     

Two-sided bounds for the asymptotic behaviour of free nonlinear vibration systems with application of the differential calculus of norms

This paper deals with the free nonlinear dynamical system [xdot](t)=A x(t)+h(t, x(t)), t≥t ₀, x(t ₀)=x ₀, where A is an n×n-matrix and h a nonlinear vector function with h(t, u)=o(∥u∥). As the first novel point, a lower bound for the asymptotic behaviour on the solution x(t) is derived. Two methods are applied to determine the optimal two-sided bounds, where one of the methods is the differential calculus of norms. In this context, the second novel point enters; it consists of a new strategy to significantly reduce the computation time for the determination of the optimal constants in the two-sided bounds. The obtained results are especially of interest in engineering and cannot be obtained by the methods used so far.     

Automata for Reduction Properties Solving

We introduce a new class of tree automata, which we call Reduction Automata (RA), and we use it to prove the decidability of the whole first order of theory of reduction (the theory of reduction is the set of true sentences built up with unary predicates red_t(x) "x is reducible by t", i.e. "some instance of t is a subterm of x"). So we link rewriting, logic and formal languages.     

Stability of θ-methods for advanced differential equations with piecewise continuous arguments

This paper deals with the stability analysis of numerical methods for the solution of advanced differential equations with piecewise continuous arguments. We focus on the behaviour of the one-leg θ-method and the linear θ-method in the solution of the equation x′(t) = ax(t + a₀x([t]) + a₁x([t+1]), with real a, a₀, a₁ and [·] designates the greatest-integer function. The stability regions of two θ-methods are determined. The conditions under which the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given.     

On a method of characteristics for solving a hyperbolic equation of second order

The hyperbolic initial-boundary value problem for the second order equationa(t,x,u)u _tt +2b(t,x,u)u _tx +c(t,x,u)u _xx =d(t,x,u,u _t ,u _x ) is solved by a special method of characteristics involving no difference equations foru _t andu _x . The discrete solution has an asymptotic expansion in even powers of the step size. Therefore, the numerical results can be improved by extrapolation to the limit.     

Adaptive robust control of nonholonomic systems with stochastic disturbances

Based on the Brockett’s necessary condition for feedback asymptotic stabilization[1], nonholonomic systems fail to be stabilized at the origin by any static continuous state feedback though they are open loop controllable. There are two novel approaches …     

Fault diagnosis in loop-connected systems

The paper considers fault diagnosis in a large system comprising a collection of small subsystems or units which can test one another for the existence of a faulty condition. If subsystem α is not faulty and tests subsystem β, a correct indication of the status of β is obtained; if α is faulty, the test outcome contains meaningless information. A particular form of interconnection is examined. For a system with n units u_o,u₁,…,u_{n ? 1}, for each i unit u_i tests u_{i + 1},u_{i + 2},…,u_{i + A} (modulo n arithmetic being understood), where A is a preselected integer. If t is the maximum number of faulty units, we show that when t ? A, all faults are immediately diagnosable if n ? 2t + 1; we also show that when t ? A, at least A faults can be diagnosed if and only if n ? s(t ? As) + t + A + 1, where s is the integer which maximizes the quadratic function f(x) = x(t ? Ax) of the integer variable x.     

A constrained optimization approach for an adaptive generalized subspace tracking algorithm

In this paper, we present an orthonormal version of the generalized signal subspace tracking. It is based on an interpretation of the generalized signal subspace as the solution of a constrained minimization task. This algorithm, referred to as the CGST algorithm, guarantees the C_x-orthonormality of the estimated generalized signal subspace basis at each iteration which C_x denotes the correlation matrix of the sequence x(t). An efficient implementation of the proposed algorithm enhances applicability of it in real time applications.     

Cylindrical decomposition for systems transcendental in the first variable

We present a generalization of the Cylindrical Algebraic Decomposition (CAD) algorithm to systems of equations and inequalities in functions of the form p(x,f₁(x),…,f_m(x),y₁,…,y_n), where p∈Q[x,t₁,…,t_m,y₁,…,y_n] and f₁(x),…,f_m(x) are real univariate functions such that there exists a real root isolation algorithm for functions from the algebra Q[x,f₁(x),…,f_m(x)]. In particular, the algorithm applies when f₁(x),…,f_m(x) are real exp-log functions or tame elementary functions.     

Optimal output-transitions for linear systems

This article addresses the optimal (minimum-input-energy) output-transition problem for linear systems. The goal is to transfer the output from an initial value (for all time t?t_i) to a final output value (for all time t?t_f). Previous methods solve this output-transition problem by transforming it into a state-transition problem; the initial and final states (x(t_i),x(t_f), respectively) are chosen and a minimum-energy state-to-state transition problem is solved. However, the choice of the initial and final states can be ad hoc and the resulting output-transition cost (input energy) may not be minimal. The contribution of this article is the solution of the optimal output-transition problem. An example system with elastic dynamics is studied to illustrate the proposed method. Simulation results are presented that show substantial reduction of transition costs with the use of the proposed method when compared to the use of minimum-energy state-to-state transitions.     

Exponential stability in mean square of neutral stochastic differential functional equations

The aim of this paper is to investigate the exponential stability in mean square for a neutral stochastic differential functional equation of the form d[x(t) − G(x_t)] = [f(t,x(t)) + g(t, x_t)]dt + σ(t, x_t)dw(t), where x_t = {x(t + s): − τ s 0}, with τ > 0, is the past history of the solution. Several interesting examples are a given for illustration.     

Adaptive stabilization of uncertain nonholonomic systems by state and output feedback

In this paper, adaptive state feedback and output feedback control strategies are presented for a class of nonholonomic systems in chained form with drift nonlinearity and parametric uncertainties. Both control laws are developed using state scaling and backstepping techniques. In particular, novel adaptive switching is proposed to overcome the uncontrollablity problem associated with x₀(t₀)=0. Observer-based output feedback design is developed when only partial system states are measurable, and a filtered observer rather than the traditional linear observer is used to handle the technical problem due to the presence of unavailable states in the regressor matrix. The proposed control strategies can steer the system globally converge to the origin, while the estimated parameters maintain bounded.     

Backstepping boundary control of Burgers’ equation with actuator dynamics

In this paper, we propose a backstepping boundary control law for Burgers’ equation with actuator dynamics. While the control law without actuator dynamics depends only on the signals u(0,t) and u(1,t), the backstepping control also depends on u_x(0,t), u_x(1,t), u_xx(0,t) and u_xx(1,t), making the regularity of the control inputs the key technical issue of the paper. With elaborate Lyapunov analysis, we prove that all these signals are sufficiently regular and the closed-loop system, including the boundary dynamics, is globally H³ stable and well posed.