Finite-dimensional Control Research Articles

Regulation of linear systems with both pointwise and distributed input delays by memoryless feedback

This paper is concerned with the stabilization of linear systems with both pointwise and distributed input delays, which can be arbitrarily large yet exactly known. The state vector used in the well-known Artstein transformation is firstly linked with the future state of the system. Pseudo-predictor feedback (PPF) approaches are then established to design memory stabilizing controllers. Necessary and sufficient conditions guaranteeing the stability of the closed-loop system are established in terms of the stability of some integral delay systems. Furthermore, since the PPF still is infinite-dimensional state feedback law and may cause difficulties in their practical implementation, truncated pseudo-predictor feedback (TPPF) approaches are established to design finite dimensional (memoryless) controllers. It is shown that the pointwise and distributed input delays can be compensated properly by the TPPF as long as the open-loop system is polynomially unstable. Finally, two numerical examples, one of which is the spacecraft rendezvous control system, are carried out to support the obtained theoretical results.

Sensor Placement for Optimal Control of Infinite-Dimensional Systems

An important challenge in controlling distributed parameter systems is implementing feedback control laws over an infinite-dimensional space. One widely studied approach is to place sensors at a set of discrete locations and then approximate the state feedback using the sensor outputs. This approach naturally raises the question of where the sensors should be located. In this paper, we investigate the problem of placing a set of sensors on the unit interval in order to minimize the mean-square deviation between a desired infinite-dimensional control law, and an approximate finite-dimensional controller obtained by applying state feedback at the chosen sensor positions. We propose a greedy algorithm and derive optimality bounds on the selected set of sensors. We also present a simplified greedy algorithm in which the incremental improvements from adding each sensor are not updated at each step. We analyze the performance of the approaches under two scenarios. In the case where the sensor placements are constrained such that each of the detection radii of each pair of sensors do not overlap, we show that the two algorithms are equivalent and achieve a 1/2 + ε optimality guarantee, where ε can be made arbitrarily small at the cost of increased computational overhead. When overlaps exist between sensor detection areas, we derive an optimality bound for the simplified algorithm. The value of the optimality bound is determined by the sensor model, cardinality of sensor set, and the allowed minimum sensor distance. Our approach is illustrated through numerical study, in which we compare our proposed greedy algorithm, the current state-of-the-art approach, and the true optimum obtained from exhaustive search.

Stabilizability of Infinite Dimensional Systems by Finite Dimensional Control

Abstract In this paper, we consider control systems for which the underlying semigroup is analytic and the resolvent of its generator is compact. In that case we give a characterization of the stabilizability of such control systems. When the stabilizability condition is satisfied, the system is also stabilizable by finite dimensional controls. We end the paper by giving an application of this result to the stabilizability of the Oseen equations with mixed boundary conditions.

Boundary stabilization of first-order hyperbolic equations with input delay

This paper is concerned with the boundary stabilization problem of first-order hyperbolic equations with input delay. In our previous work, the same problem for parabolic equations with input delay was addressed. For the hyperbolic equation, the element of time lag is similarly replaced by a transport equation and a backstepping method is employed. However, unlike in the case of the parabolic equation, it is difficult to obtain the eigenvalues and eigenfunctions of the system operator for the hyperbolic equation. Hence, we cannot construct the target system and the controller by using the finite-dimensional control theory together. In this paper, using $$C_0$$ -semigroups for hyperbolic equations with nonlocal boundary condition, we show that the proposed controller is expressed by an abstract form in a Hilbert space so-called a predictor, and that the predictor makes sense under a condition. Further, for the inverse integral transformation, we obtain an interesting result on its continuity under the same condition. A numerical algorithm is also proposed for solving a non-standard hyperbolic equation appearing in our controller design.

An optimal control theory for nonlinear optimization

The Karush–Kuhn–Tucker conditions for a given nonlinear programming problem are generated as the transversality conditions of an optimal control problem. The directional derivatives of the objective- and constraint-functions supply the vector fields for the optimal control problem with the search vector as the control variable. Zero-Hamiltonian trajectories along the steepest descent of control Lyapunov functions provide optimal optimization algorithms. The optimality of the algorithm also depends upon the choice of a metric for the finite dimensional control space. Many well-known algorithms – such as Newton’s method, the first-order Lagrangian method, the steepest descent method and Richardson’s method, to name a few – are derived by minimizing the Lie derivative of a quadratic control Lyapunov function. Merit functions in optimization may also be generated using the concept of a control Lyapunov function. These results suggest that optimal control principles hold the potential for a unified theory for optimization.

Multiobjective model predictive control for stabilizing cost criteria

In this paper we demonstrate how multiobjective optimal control problems can be solved by means of model predictive control. For our analysis we restrict ourselves to finite-dimensional control systems in discrete time. We show that convergence of the MPC closed-loop trajectory as well as upper bounds on the closed-loop performance for all objectives can be established if the ‘right’ Pareto-optimal control sequence is chosen in the iterations. It turns out that approximating the whole Pareto front is not necessary for that choice. Moreover, we provide statements on the relation of the MPC performance to the values of Pareto-optimal solutions on the infinite horizon, i.e. we investigate on the inifinite-horizon optimality of our MPC controller.

Boundary stabilization of a fluid-rigid body interaction system

Abstract Let us consider a fluid-rigid body interaction system. We are interested in the feedback stabilization of this system by using a finite-dimensional control localized on the interface between the structure and the fluid. The fluid is assumed to be viscous and incompressible and to follow the Navier-Stokes system and we consider for the rigid body the Newton laws. We follow a general method for the stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domain is moving with time. We prove that for small initial velocities and if the initial position and the final position are close, we can stabilize the position and the velocity of the rigid body and the velocity of the fluid.

Late-lumping backstepping control of partial differential equations

We consider in this paper three different partial differential equations (PDEs) that can be exponentially stabilized using backstepping controllers. For implementation, a finite-dimensional controller is generally needed. The backstepping controllers are approximated and it is proven that the finite-dimensional approximated controller stabilizes the original system if the order is high enough. This approach is known as late-lumping. The other approach to controller design for PDEs first approximates the PDE and then a controller is designed; this is known as early-lumping. Simulation results comparing the performance of late-lumping and early-lumping controllers are provided.

Finite‐dimensional guaranteed cost sampled‐data fuzzy control of Markov jump distributed parameter systems via T–S fuzzy model

In this study, the authors are concerned with a finite-dimensional guaranteed cost sampled-data fuzzy control problem, which is addressed for a class of non-linear Markov jump parabolic partial differential equation (MJPPDE) systems. Applying the Galerkin's method to the MJPPDE system, a non-linear Markov jump ordinary differential equation (MJODE) system is derived. The resulting non-linear MJODE system can describe the dominant dynamics of the MJPPDE system, which is subsequently expressed by the Takagi–Sugeno (T–S) fuzzy model. Then, a guaranteed cost sampled-data fuzzy controller is developed to stabilise the closed-loop slow fuzzy system exponentially in the mean-square sense while providing an upper bound for the quadratic cost function. Making full use of their special features, they will be able to establish the bound on . The authors' theory enables us to guarantee the existence of the feedback controller and to judge the effectiveness of the feedback control based on the sampled-data to stabilise the given system in the sense of mean-square exponential stability.

A Novel Discrete-time Nonlinear Model Predictive Control Based on State Space Model

This paper proposes a novel finite dimensional discrete-time Nonlinear Model Predictive Control. This technique is based on discrete-time state-space models, Taylor series expansion for prediction and performance index optimization. Furthermore, the technique extends the concept of the Lie derivative for the discrete time case using Euler backwards method. The performance validation for the discrete-time Nonlinear Model Predictive Control uses the simulation of a single-link flexible joint robot and the inverted pendulum. Comparison of the proposed finite dimensional discrete-time Nonlinear Model Predictive Control technique with Feedback Linearization Control is also discussed. Analytical and numerical results show excellent performances for both, the single-link flexible joint and inverted pendulum controllers using the proposed discrete-time Nonlinear Model Predictive Control technique.

Minimal controllability time for finite-dimensional control systems under state constraints

We consider the controllability problem for finite-dimensional linear autonomous control systems, under state constraints but without imposing any control constraint. It is well known that, under the classical Kalman condition, in the absence of constraints on the state and the control, one can drive the system from any initial state to any final one in an arbitrarily small time. Furthermore, it is also well known that there is a positive minimal time in the presence of compact control constraints. We prove that, surprisingly, a positive minimal time may be required as well under state constraints, even if one does not impose any restriction on the control. This may even occur when the state constraints are unilateral, like the nonnegativity of some components of the state, for instance. Using the Brunovsky normal forms of controllable systems, we analyze this phenomenon in detail, that we illustrate by several examples. We discuss some extensions to nonlinear control systems and formulate some challenging open problems.

Finite-Approximate Controllability of Fractional Evolution Equations: Variational Approach

Abstract In this work we extend a variational method to study the approximate controllability and finite dimensional exact controllability (finite-approximate controllability) for the fractional semilinear evolution equations with nonlocal conditions in Hilbert spaces. Assuming the approximate controllability of the corresponding linear equation we obtain sufficient conditions for the finite-approximate controllability of the fractional semilinear evolution equation under natural conditions. The obtained results are generalization and continuation of the recent results on this issue. Applications to heat equations are treated.

Stabilizability of Infinite-Dimensional Systems by Finite-Dimensional Controls

Abstract In this paper, we consider control systems for which the underlying semigroup is analytic, and the resolvent of its generator is compact. In that case we give a characterization of the stabilizability of such control systems. When the stabilizability condition is satisfied the system is also stabilizable by finite-dimensional controls. We end the paper by giving an application of this result to the stabilizability of the Oseen equations with mixed boundary conditions.

Stable and Robust Controller Synthesis for Unstable Time Delay Systems via Interpolation and Approximation

In this paper, we study the robust stabilization of a class of single input single output (SISO) unstable time delay systems by stable and finite dimensional controllers through finite dimensional approximation of infinite dimensional parts of the plant. The plant of interest is assumed to have finitely many non-minimum phase zeros but is allowed to have infinitely many unstable poles in the open right half plane. Conservatism of the proposed methods is illustrated by numerical examples for which infinite dimensional strongly stabilizing controllers are derived in the literature.

Adaptive H Consensus Control for Distributed Parameter Systems of Hyperbolic Type on Directed Graph

Abstract A design method of adaptive H∞ consensus control of multi-agent systems composed of a class of infinite-dimensional systems on directed network graphs, is presented in this paper. The proposed control scheme is derived as a solution of certain H∞ control problems, where the effects of neglected infinite-dimensional modes are regarded as external disturbances to the process. It is shown that the desirable consensus tracking is achieved approximately via the finite dimensional adaptive controller.

Turnpike Properties and Strict Dissipativity for Discrete Time Linear Quadratic Optimal Control Problems

We analyze the connection between turnpike behaviors and strict dissipativity properties for discrete time finite dimensional linear quadratic optimal control problems. We first use strict dissipativity as a sufficient condition for the turnpike property. Next, we characterize strict dissipativity and the newly introduced property of strict pre-dissipativity in terms of the system matrices related to the linear quadratic problem. These characterizations then lead to new necessary conditions for the turnpike properties under consideration, and thus eventually to necessary and sufficient conditions in terms of spectral criteria and matrix inequalities. One of the key novelties which distinguishes the results in the present paper from earlier ones on linear quadratic optimal control problems is the consideration of state and input constraints.

On the explicit scheme with variable time steps for solving the parabolic optimal control problem

The optimal control problem including a linear parabolic equation as the state problem is considered. Pointwise constraints are imposed on the control function. The objective functional contains a given observation function on the entire domain at each moment of time. The optimal control problem is approximated by a finite-dimensional problem with grid approximation of the state equation by using an explicit scheme with variable time steps. The existence and uniqueness of solutions for the continuous and grid optimal control problems are proved. The finite-dimensional optimal control problemis solved by the Udzawa iterationmethod. Results of numerical experiments are presented.

Tracking of periodic reference signal: A parameterized finite dimensional repetitive control approach

This paper is concerned with the development of a Modified Finite Dimensional Repetitive Control (MFDRC) system. Conventional FDRC is modified through parameterization of the controller using co-prime factorization. The proposed method of controller design specifies the input output characteristics beforehand, ensuring highly accurate tracking and disturbance rejection property of the system subjected to periodic reference input. Another problem is the stabilization issues in the input–output and the disturbance rejection characteristics of a conventional repetitive control system with respect to the periodic reference input owing to have infinite number of poles in the transfer function of it. It is desirable that the transfer functions from both the reference input and the disturbance to the output have a finite number of poles and that will be taken care of in the proposed scheme of MFDRC design depicted in this paper. The performance of designed controller has been validated in steps applied to a laboratory based servo system. Comparisons are done using conventional PID controller, FDRC-based PID controller and MFDRC to show advantage, disadvantage or limitation of applied control scheme.

Control scheme for LTI systems with Lipschitz non‐linearity and unknown time‐varying input delay

In this study, the authors propose a control structure for a class of linear time-invariant (LTI) systems with Lipschitz non-linearity and unknown time-varying input delay. This scheme considers the worst-case scenario in control design with truncated prediction feedback approach, and takes into account the information of the lower bound of delay in the stability analysis. A finite-dimensional controller is constructed, requiring neither the non-linear function nor the exact delay function. The truncated prediction deviation is minimised by employing the delay range, and then bounded by integral construction and related techniques. Within the framework of Lyapunov–Krasovskii functionals, sufficient delay–range-dependent conditions are derived for the closed-loop system to guarantee the global stability. Two numerical examples are given to validate the proposed control design.

Globally Stabilizing Finite-Dimensional Damping Control for Dissipative PDE with Bounded Inputs

The main aim of this paper is to address the global asymptotic stabilization (GAS) of a class of dissipative partial differential equations (pdes) via finite-dimensional admissible (regular and bounded) damping (output) controls. To this end, we use the inertial manifold theory to derive infinite-dimensional dissipative control systems given by interconnection of a finite-dimensional control system on the inertial manifold plus an infinite-dimensional zero-input system on the complement. We show that the GAS of such systems is reduced to the finite-dimensional one. Then, we prove that the finite-dimensional systems which are zero-input point-dissipative (have global attractors K) and those which are B-strictly passive (passivity relative to bounded sets) are connected. Finally, we use the control Lyapunov functions (CLF) theory to design admissible feedback damping controls for the GAS of B-strictly passive systems.