Finite-dimensional Control Research Articles

Optimal finite-dimensional controller of the stochastic differential object’s state by its output II. Stochastic measurements and separation theorem

Consideration is continued of the problem of the inertial control law by the output synthesis of a continuous nonlinear stochastic plant, which is optimal on average and on a finite time interval, and works with the desired speed. An algorithm for synthesizing the optimal structure of a dynamic controller of a selected finite order, obtained in the first part of the article for the case of accurate measurements of a of the control object’s state variables part, is presented. Its application is demonstrated in detail for the case when the state variables of an object are measured with random errors. Using the example of a linear-quadratic-Gaussian problem, it is shown that the proposed controller of the corresponding order also satisfies the well-known separation theorem.

Optimal Finite-Dimensional Controller of a Stochastic Differential Object’s State by Its Output. II. Stochastic Measurements and Separation Theorem

Optimal Finite-Dimensional Controller of a Stochastic Differential Object’s State by Its Output. II. Stochastic Measurements and Separation Theorem

Reduced order in domain control of distributed parameter port-Hamiltonian systems via energy shaping

An in-domain finite dimensional controller for a class of distributed parameter systems on a one-dimensional spatial domain formulated under the port-Hamiltonian framework is presented. Based on (Trenchant et al. 2017) where positive feedback and a late lumping approach is used, we extend the Control by Interconnection method and propose a new energy shaping methodology with an early lumping approach on the distributed spatial domain of the system. Our two main control objectives are to stabilize the closed-loop system, as well as to improve the closed-loop dynamic performances. With the early lumping approach, we investigate two cases of the controller design, the ideal case where each distributed controller acts independently on the spatial domain (fully-actuated), and the more realistic case where the control action is piecewise constant over certain intervals (under-actuated). We then analyze the asymptotic stability of the closed-loop system when the infinite dimensional plant system is connected with the finite dimensional controller. Furthermore we provide simulation results comparing the performance of the fully-actuated case and the under-actuated case with an example of an elastic vibrating string.

Cost evaluation of finite dimensional and approximate null-controllability for the one dimensional half-heat equation

In this article we evaluate the cost of the finite dimensional and the approximate controllability of a one dimensional anomalous diffusion equation which is not null-controllable. The finite dimensional controllability property consists in driving to zero a finite number of frequencies of the corresponding solutions. To evaluate its cost, we construct an explicit biorthogonal sequence to a finite part of a family of real exponential functions. Since the exponents of these functions satisfy the so-called Müntz density condition, there is no biorthogonal sequence to the entire family which shows that the equation is not even spectrally controllable. For the approximate controllability problem, we combine the idea of finite dimensional controllability studied before and the intrinsic dissipative nature of the system. These arguments allow to show that the controlled solution can be made arbitrarily small after sufficiently large time and provide estimates for the corresponding approximate controls.

Finite-Approximate Controllability of ν-Caputo Fractional Systems

This paper introduces a methodology for examining finite-approximate controllability in Hilbert spaces for linear/semilinear ν-Caputo fractional evolution equations. A novel criterion for achieving finite-approximate controllability in linear ν-Caputo fractional evolution equations is established, utilizing resolvent-like operators. Additionally, we identify a control strategy that not only satisfies the approximative controllability property but also ensures exact finite-dimensional controllability. Leveraging the approximative controllability of the corresponding linear ν-Caputo fractional evolution system, we establish sufficient conditions for achieving finite-approximative controllability in the semilinear ν-Caputo fractional evolution equation. These findings extend and build upon recent advancements in this field. The paper also explores applications to ν-Caputo fractional heat equations.

Optimal Finite-Dimensional Controller of the Stochastic Differential Object’s State by Its Output. I. Incomplete Precise Measurements

Optimal Finite-Dimensional Controller of the Stochastic Differential Object’s State by Its Output. I. Incomplete Precise Measurements

A Finite-Dimensional Control Scheme for Fractional-Order Systems under Denial-of-Service Attacks

In this article, the security control problem of discrete-time fractional-order networked systems under denial-of-service (DoS) attacks is considered. A practically applicable finite-dimensional control strategy will be developed for fractional-order systems that possess nonlocal characteristics. By employing the Lyapunov method, it is theoretically proved that under the proposed controller, the obtained closed-loop fractional system is globally input-to-state stable (ISS), even in the presence of DoS attacks. Finally, the effectiveness of the designed control method is demonstrated by the numerical example.

Optimal Finite-Dimensional Controller of the Stochastic Differential Object’s State by Its Output. I. Incomplete Precise Measurements

The well-known problem of synthesizing the optimal on the average and on given time interval of the inertial control law for a continuous stochastic object if only a part of its state variables are accurately measured is considered. Due to the practical unrealizability of its classical infinite-dimensional Stratonovich-Mortensen solution, it is proposed to limit ourselves to optimizing the structure of a finite-dimensional dynamic controller, whose order is chosen by the user. This finiteness allows using a truncated version of the a posteriori probability density that satisfies a deterministic partial differential integrodifferential equation. Using the Krotov extension principle, sufficient optimality conditions for the structural functions of the controller and the Lagrange–Pontryagin equation for finding their extremals are obtained. It is shown that in particular cases of the absence of measurements, complete measurements and taking into account only the values of incomplete measurements, the proposed controller turns out to be static (inertialess), and the relations for its synthesis coincide with the known ones. For a dynamic controller, algorithms for finding each of its structural functions are given.

On Stackelberg equilibrium in the sense of program strategies in Volterra functional operator games

For a nonlinear Volterra functional operator equation controlled by two players with the help of finite dimensional program controls with integral objective functionals we prove existence of Stackelberg equilibrium (in the style of M.S.Nikol'skiy). On this way we use our formerly proved results on continuous dependence of the state and functionals on finite dimensional controls and also classical Weierstrass theorem. The property of being singleton for the minimizer set of the first player is proved by the scheme of M.S. Nikol'skiy applied earlier for a linear ordinary differential equation.

Finite dimensional shape control design of linear port-Hamiltonian systems with in-domain pointwise inputs

This paper is concerned with shape control of a class of infinite-dimensional port-Hamiltonian system, using an early lumping approach, i.e. the dynamic controller synthesized from a low-order discretized version of the system. The approach provides an optimal criterion for choosing a free parameter of the controller so that the closed-loop system converges to the best approximation of the desired imposed shape. The methodology is based on the so called control by interconnection method and structural invariant from which an analytical expression is obtained for the shapes that the system can actually achieve. An Euler-Bernoulli beam model with pointwise inputs is used as example to illustrate the proposed methodology.

Local Output Feedback Stabilization of a Reaction–Diffusion Equation With Saturated Actuation

This article is concerned with the output feedback stabilization of a reaction–diffusion equation by means of bounded control inputs in the presence of saturations. Using a finite-dimensional controller composed of an observer coupled with a finite-dimensional state-feedback, we derive a set of conditions ensuring the stability of the closed-loop plant while estimating the associated domain of attraction in the presence of saturations. This set of conditions is shown to be always feasible for an order of the observer selected large enough. The stability analysis relies on Lyapunov functionals along with a generalized sector condition classically used to study the stability of linear finite-dimensional plants in the presence of saturations.

OUTPUT CONTROLLABILITY OF DELAYED CONTROL SYSTEMS IN A LONG TIME HORIZON

In this paper, we consider the output controllability of finite-dimensional control systems governed by a distributed delayed control. For systems with ordinary controls, this problem was investigated earlier. Nevertheless, in many practical and technical problems the control acts with some delay. We give the necessary and sufficient condition for the output controllability. The main goal of our control is to govern the output of the system to some position on a subspace in a given instant, and then keep this output fixed for the remaining times. This property is called the long-time output controllability. For this, sufficient conditions are given. The introduced notions are applied for the investigation of averaged controllability of systems with delayed controls. The general approach for that is to approximate the system by the ordinary ones. Some examples are considered.

Error estimates for halfspace regularization of state constrained multiobjective elliptic control problems

This paper generalizes the halfspace regularization given in (Jadamba et al. in Syst Control Lett 61(6):707–713, 2012) from scalar to multiobjective optimization for an abstract linearly constrained multiobjective optimization problem. We establish quantitative error estimates for the regularization error in terms of the Hausdorff distance of the efficient set of the original problem to the regularized efficient set. We apply these results to state-constrained multiobjective optimal control problems. In particular, we consider two different situations: distributed and finite-dimensional controls, and for each one, we define two different halfspace regularization schemes given in terms of a family of triangulations of the domain. This leads to two regularized problems corresponding to finitely many pointwise and integral constraints. By applying the abstract results, we get regularization error estimates for the four control problems under general conditions.

Robust output regulation of a thermoelastic system

In this paper, we consider the robust output regulation problem for the thermoelastic system where the boundary control input and the output are located at the left end of the wave equation and the disturbance input is located at the right end of the heat equation. We formulate the system as a boundary control system and prove that it is impedance passive and well-posed. We design a finite-dimensional controller based on the internal model principle and show that the controller can achieve robust output tracking and disturbance rejection. The numerical simulations demonstrate that the finite-dimensional dynamic error feedback controller can make the output track the reference signal asymptotically and the state of the closed-loop system is bounded.

Efficient scalarization in multiobjective optimal control of a nonsmooth PDE

This work deals with the efficient numerical characterization of Pareto stationary fronts for multiobjective optimal control problems with a moderate number of cost functionals and a mildly nonsmooth, elliptic, semilinear PDE-constraint. When “ample” controls are considered, strong stationarity conditions that can be used to numerically characterize the Pareto stationary fronts are known for our problem. We show that for finite dimensional controls, a sufficient adjoint-based stationarity system remains obtainable. It turns out that these stationarity conditions remain useful when numerically characterizing the fronts, because they correspond to strong stationarity systems for problems obtained by application of weighted-sum and reference point techniques to the multiobjective problem. We compare the performance of both scalarization techniques using quantifiable measures for the approximation quality. The subproblems of either method are solved with a line-search globalized pseudo-semismooth Newton method that appears to remove the degenerate behavior of the local version of the method employed previously. We apply a matrix-free, iterative approach to deal with the memory and complexity requirements when solving the subproblems of the reference point method and compare several preconditioning approaches.

Interior and H∞ feedback stabilization for sabra shell model of turbulence

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models are studied for their mathematical relevance and the numerical simulations that exhibit at most resemblance with turbulent flows. One of the mathematically well‐studied shell model of turbulence is called sabra shell model. This work concerns with two important issues related to shell model, namely, feedback stabilization and robust stabilization. We first address stabilization problem related to sabra shell model of turbulence and prove that the system can be stabilized via finite dimensional controller. Thus, only finitely many modes of the shell model would suffice to stabilize the system. Later, we study robust stabilization in the presence of the unknown disturbance and corresponding control problem by solving an infinite time horizon max‐min control problem. We first prove the stabilization of the associated linearized system and characterize the optimal control in terms of a feedback operator by solving an algebraic Riccati equation. Using the same Riccati operator, we establish asymptotic stability of the nonlinear system.

Output feedback stabilization of reaction–diffusion PDEs with a non-collocated boundary condition

This paper addresses the control design problem of output feedback stabilization of a reaction–diffusion PDE with a non-collocated boundary condition. More precisely, we consider a reaction–diffusion equation with a boundary condition describing a proportional relationship between the left and right Dirichlet traces. Such a boundary condition naturally emerges, e.g., in the context of reaction–diffusion partial differential equations presenting a transport term and with a periodic Dirichlet boundary condition. The control input takes the form of the left Neumann trace. Finally, the measurement is selected as a pointwise Dirichlet measurement located either in the domain or at the boundary. The adopted control strategy takes the form of a finite-dimensional controller coupling a state feedback and a finite-dimensional observer. The stability of the closed-loop system is obtained provided the order of the observer is selected to be large enough. Finally, we extend this result to the establishment of an input-to-state stability estimate with respect to an additive perturbation in the application of the boundary control.

Delay-based stabilisation and strong stabilisation of LTI systems by nonsmooth constrained optimisation

Stabilisation and strong stabilisation (i.e. the problem of stabilising by a stable controller) of linear time-invariant (LTI) multi-input multi-output systems are considered. The considered systems may be finite-dimensional or time-delay systems, which may have multiple discrete time delays. The controllers to be designed may also be either finite-dimensional or time-delay LTI controllers. In the controller design phase, a constrained nonsmooth optimisation-based design method is employed. The effectiveness of the proposed method is illustrated through various benchmark examples which are solved by using the recently developed Matlab-based software called DrStabilization . The examples also show that, in many cases, time-delay controllers may have certain advantages over conventional finite-dimensional controllers.

Computation of open-loop inputs for uniformly ensemble controllable systems

<p style='text-indent:20px;'>This paper presents computational methods for families of linear systems depending on a parameter. Such a family is called ensemble controllable if for any family of parameter-dependent target states and any neighborhood of it there is a parameter-independent input steering the origin into the neighborhood. Assuming that a family of systems is ensemble controllable we present methods to construct suitable open-loop input functions. Our approach to solve this infinite-dimensional task is based on a combination of methods from the theory of linear integral equations and finite-dimensional control theory.</p>

Finite-Dimensional Controllers for Consensus in a Leader-Follower Network of Marginally Unstable Infinite-Dimensional Agents

We consider the output-feedback state consensus problem for a homogeneous multi-agent system consisting of one leader agent and N follower agents. The dynamics of each of these agents is governed by a single-input single-output regular linear system (RLS), with the input to the leader agent being zero at all times. The transfer function of this RLS has all its poles in the closed left-half of the complex-plane, with only a finite number of them lying on the imaginary axis. Each follower agent can access the relative output of all its neighboring agents and some of the follower agents can access the relative output of the leader. The communication graph associated with the exchange of relative outputs is directed. Under this setting, we first establish that a controller solves the above leader-follower consensus problem if and only if it can simultaneously stabilize N regular linear systems. We then adapt a recently developed frequency-domain technique to construct a stable finite-dimensional output-feedback controller which solves the simultaneous stabilization problem, and hence the consensus problem. We demonstrate the efficacy of our controller design technique using a numerical example.