Discrete-time State-space Systems Research Articles

On positive realness and negative imaginariness of uncertain discrete-time state-space symmetric systems

ABSTRACTThis paper provides an analysis of uncertain discrete-time negative imaginary and positive real systems with state-space symmetry. First, a necessary and sufficient condition is provided for a minimal state-space symmetric system to be discrete-time positive real or discrete-time negative imaginary. Then, based on the discrete-time symmetric negative imaginary and positive real lemmas, bounds on symmetrically structured perturbations that maintain the negative imaginariness and positive realness of a discrete-time state-space symmetric system are derived. Finally, an RL network example is provided to illustrate the main results of the paper.

Perfect Control for Right-Invertible Grünwald-Letnikov Plants–an Innovative Approach to Practical Implementation

Abstract A new perfect control algorithm dedicated to fractional-order right-invertible systems, i.e. plants with a greater number of input than output variables, is presented in this paper. It is shown that such a control strategy can be particularly applied with regard to practical tasks. Henceforth, the Grünwald-Letnikov difference operator Δ α of an assumed order α can be truncated without loss of generality. For that reason, the so-called pole-free perfect control formula can be used to minimize the essential drawback of the Grünwald-Letnikov approach engaged, so as to define the intriguing issue regarding the robust perfect control for non-integer-order right-invertible LTI discrete-time state-space systems. Simulation examples show that the presented method can compete with a classical stable-pole one, for which the actual systems described by a fractional-order model often correspond with an inconvenient asymptotic perfect control solution given by the unlimited original operator Δ α . In the end, the possibility of employing of author’s nonunique right inverses dedicated to nonsquare MIMO system matrices is demonstrated, thus giving rise to the introduction of a new powerful tool for robustification of non-integer-order closed-loop perfect control plants as well.

Stability Analysis of the Bat Algorithm Described as a Stochastic Discrete-Time State-Space System

The main problem with the soft-computing algorithms is a determination of their parameters. The tuning rules are very general and need experiments during a trial and error method. The equations describing the bat algorithm have the form of difference equations, and the algorithm can be treated as a stochastic discrete-time system. The behaviour of this system depends on its dynamic and preservation stability conditions. The paper presents the stability analysis of the bat algorithm described as a stochastic discrete-time state-space system. The observability and controllability analyses were made in order to verify the correctness of the model describing the dynamic of BA. Sufficient conditions for stability are derived based on the Lyapunov stability theory. They indicate the recommended areas of the location of the parameters. The analysis of the position of eigenvalues of the state matrix shows how the different values of parameters affect the behaviour of the algorithm. They indicate the recommended area of the location of the parameters. Simulation results confirm the theory-based analysis.

Finite memory controls for discrete-time state-space systems

As a new type of output feedback control for a stochastic discrete-time state-space system, a finite memory control (FMC) is proposed and its properties are investigated. Instead of using an internal state involved with old data, the FMC is required to have finite memory with respect to inputs and outputs together with the future reference signals and minimize a conditional receding horizon quadratic performance criterion. It turns out that the FMC is linearly expressed in terms of inputs and outputs on the recent time, and the future reference signals on the finite time. As the linear quadratic Gaussian (LQG) controls with infinite memory structure can be separated into the linear quadratic (LQ) controls and the Kalman filters, the proposed FMC consists of a receding horizon control (RHC) for state feedback control and a minimum variance finite impulse response (MVFIR) filter for state estimation. It is shown that closed loop poles are made up of those involved with the RHC and the rest of zeros. The stability is also shown to be guaranteed under some condition.

A mathematical model of a slab reheating furnace with radiative heat transfer and non-participating gaseous media

A mathematical model of the reheating process of steel slabs in industrial fuel-fired furnaces is developed. The transient temperature field inside the slabs is computed by means of the Galerkin method. Radiative heat transfer inside the furnace constitutes boundary conditions that couple the dynamic subsystems of the slabs. Constraining the heat fluxes to piecewise linear, discontinuous signals furnishes a discrete-time state-space system. Conditions for an exponential decrease of the open-loop control error are derived. Measurements from an instrumented slab in the real system demonstrate the accuracy of the model. The simple and computationally inexpensive model is suitable for trajectory planning, optimization, and controller design.

Digital Modeling and PID Controller Design for MIMO Analog Systems with Multiple Delays in States, Inputs and Outputs

This paper presents a discrete-time state-space methodology for the digital modeling and design of an optimal digital proportional-integral-derivative (PID) plus state-feedback controller for multiple-input, multiple-output (MIMO) continuous-time systems with multiple time delays in states, inputs and outputs. To implement the digital design, first the Chebyshev quadrature formula together with a linear interpolation method is employed to obtain an extended discrete-time state-space model from the continuous-time multiple time-delay system. Then, a partially predetermined digital PID controller and the extended discrete-time state-space model are formulated as an augmented discrete-time state-space system utilizing state-feedforward and state-feedback linear-quadratic regulator (LQR) design. As a result, the parameters of the optimal PID controller and its associated state-feedback controller can be determined by tuning the weighting matrices in the LQR performance criteria. Further, an optimal discrete-time observer is jointly constructed for the multivariable system with multiple delays in states, inputs and outputs. The proposed design methodology can be applied to general MIMO continuous-time multiple time-delay systems for performance improvement and disturbance rejection. An illustrative example is given to demonstrate the effectiveness of the developed method.

Gramian-Preserving Frequency Transformation for Linear Discrete-Time State-Space Systems

This paper proposes the Gramian-preserving frequency transformation for linear discrete-time state-space systems. In this frequency transformation, we replace each delay element of a discrete-time system with an allpass system that has a balanced realization. This approach can generate transformed systems that have the same controllability/observability Gramians as those of the original system. From this result, we show that the Gramian-preserving frequency transformation gives us transformed systems with different magnitude characteristics, but with the same structural property with respect to the Gramians as that of the original system. This paper also presents a simple method for realization of the Gramian-preserving frequency transformation. This method makes use of the cascaded normalized lattice structure of allpass systems.

<formula formulatype="inline"><tex>${\cal H}_{\infty}$</tex> </formula> Finite Memory Controls for Linear Discrete-Time State-Space Models

In this brief, we propose a new type of output feedback control, called a Hinfin finite memory control (HFMC), for discrete-time state-space systems. Some constraints, such as linearity, unbiasedness to the optimal state feedback control, and finite memory structure with respect to an input and an output are required in advance. Among the controls needed with these requirements, we choose the HFMC to optimize the Hinfin performance criterion. The HFMC is obtained by solving the linear matrix inequality problem with a parameterization of a linear equality constraint. We show through simulation that the HFMC is more robust against uncertainties, and is faster in convergence than existing Hinfin output feedback controls

Stabilization and [formula omitted] control of discrete-time symmetric systems

We examine the H ∞ control analysis, the output feedback stabilization, and the output feedback H ∞ control synthesis problems for discrete-time state-space symmetric systems. We obtain explicit analytical solutions for the H ∞ norm of such systems, and an explicit parametrization of the output feedback stabilizing controllers and H ∞ controllers. Extensions to robust and positive real control of such systems are also examined. These results are obtained from the linear matrix inequality formulations of the stabilization and the H ∞ control synthesis problems for discrete-time systems using simple matrix algebraic tools. The results have obvious computational advantages, especially for large scale symmetric systems.

A characterization of solutions of the discrete-time algebraic Riccati equation based on quadratic difference forms

This paper is concerned with a characterization of all symmetric solutions to the discrete-time algebraic Riccati equation (DARE). Dissipation theory and quadratic difference forms from the behavioral approach play a central role in this paper. Along the line of the continuous-time results due to Trentelman and Rapisarda [H.L. Trentelman, P. Rapisarda, Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation, SIAM J. Contr. Optim. 40 (3) (2001) 969–991], we show that the solvability of the DARE is equivalent to a certain dissipativity of the associated discrete-time state space system. As a main result, we characterize all unmixed solutions of the DARE using the Pick matrix obtained from the quadratic difference forms. This characterization leads to a necessary and sufficient condition for the existence of a non-negative definite solution. It should be noted that, when we study the DARE and the dissipativity of the discrete-time system, there exist two difficulties which are not seen in the continuous-time case. One is the existence of a storage function which is not a quadratic function of state. Another is the cancellation between the zero and infinite singularities of the dipolynomial spectral matrix associated with the DARE, due to the infinite generalized eigenvalues of the associated Hamiltonian pencil. One of the main contributions of this paper is to demonstrate how to resolve these difficulties.

RECEDING HORIZON FINITE MEMORY OUTPUT TRACKING CONTROL AND ITS APPLICATION TO INVERTED PENDULUM CONTROL

In this paper, a new type of output feedback tracking control, called a receding horizon finite memory output tracking control (RHFMOTC), is proposed for discrete-time state-space systems. Constraints such as linearity and finite memory structure with respect to inputs/outputs and future output references, and unbiasedness from the optimal state feedback tracking control are required in advance. The RHFMOTC is chosen to minimize a receding horizon LQG tracking performance criterion with these constraints. The stability and the separation principle of the RHFMOTC are investigated. It is shown through application to inverted pendulum control that the RHFMOTC has the better tracking performance than the linear quadratic Gaussian tracking control (LQGTC).

RECEDING HORIZON FINITE MEMORY CONTROLS FOR OUTPUT FEEDBACK CONTROLS OF DISCRETE-TIME STATE SPACE SYSTEMS

In this paper, a new type of output feedback control, called a receding horizon finite memory control (RHFMC), is proposed for stochastic discrete-time state space systems. Constraints such as linearity and finite memory structure with respect to an input and an output, and unbiasedness from the optimal state feedback control are required in advance. The proposed RHFMC is chosen to minimize an optimal criterion with these constraints. The RHFMC is obtained in an explicit closed form using the output and input information on the recent time interval. It is shown that the RHFMC consists of a receding horizon control and an FIR filter. The stability of the RHFMC is investigated for stochastic systems.

THE LINEAR QUADRATIC DYNAMIC GAME FOR DISCRETE-TIME DESCRIPTOR SYSTEMS

The linear quadratic zero-sum dynamic game for discrete time descriptor systems is considered. A method, which involves solving a linear quadratic zero-sum dynamic game for a reduced-order discrete time state space system, is developed to find the linear feedback saddle-point solutions of the problem. Checkable conditions, which are described in terms of two dual algebraic Riccati equations and a Hamiltonian matrix, are given such that the linear quadratic zero-sum dynamic game for the reduced-order discrete time state space system is available. Sufficient conditions for the existence of the solutions are obtained. In contrast with the dynamic game in state space systems, the dynamic game in descriptor systems admits uncountably many linear feedback saddle-point solutions. All these solutions have the same existence conditions and achieve the same value of the dynamic game.

Performance assessment and benchmarking LQG predictive optimal controllers for discrete-time state-space systems

The performance assessment and benchmarking of discrete-time multivariable LQG predictive optimal control problems is considered for systems represented in state equation form. The class of predictive controllers represents the most popular multivariable design methods for the process industries. It is claimed that these methods provide improved performance but the question addressed is how this performance should be judged. A multistep LQGPC optimal control cost-function is minimized where future set-point or reference knowledge is assumed. The predictive control cost-function includes the future tracking error and control signal components. The state-equation system description can be written in terms of these future inputs, so that the model includes the outputs for time t and a vector of future outputs. The benchmark cost values are obtained from the solution of appropriate Riccati and Lyapunov equations. The results throw new light on the relationship between predictive, LQ and LQG control laws and more importantly into the way the performance of predictive controls should be assessed.

State space calculations for discrete probability densities

State space methods have proved to be powerful theoretical and computational tools in a number of areas of applications, in particular filtering and control theory. In this paper we advocate the use of state space methods for the study of discrete probability densities on the set {0,1,2,…}. The fundamental approach is to consider the class D of all discrete probability densities that can be represented as the impulse response/convolution kernel of a finite dimensional discrete time state space system. We show that all standard operations such as the calculation of moments, convolution, scaling, translation, product, etc. can be carried out using system representations.

Some facts about the choice of the weighting matrices in Larimore type of subspace algorithms

In this paper the effect of some weighting matrices on the asymptotic variance of the estimates of linear discrete time state space systems estimated using subspace methods is investigated. The analysis deals with systems with white or without observed inputs and refers to the Larimore type of subspace procedures. The main result expresses the asymptotic variance of the system matrix estimates in canonical form as a function of some of the user choices, clarifying the question on how to choose them optimally. It is shown, that the CCA weighting scheme leads to optimal accuracy. The expressions for the asymptotic variance can be implemented more efficiently as compared to the ones previously published.

Global numerical approach to constrained nonlinear generalized predictive control

Abstract A global numerical approach is proposed to solve the nonlinear generalized predictive control (GPC) law. Set inversion via interval analysis (SIVIA) is used to approximate the input sequences with a given length that drive a nonlinear discrete-time state-space system from a given initial state to a given terminal state in a guaranteed way. No requirement other than computability is put on the state equation and objective function.

Global numerical approach to nonlinear discrete-time control

Interval analysis is used to characterize the set of all input sequences with a given length that drive a nonlinear discrete-time state-space system from a given initial state to a given set of terminal states. No requirement other than computability (i.e. ability to be evaluated by a finite algorithm) is put on the nature of the state equations. The method is based on an algorithm for set inversion and approximates the solution set in a guaranteed way.

Expansion of det(A+B+C) and robustness analysis of discrete-time state-space systems

In this paper, the expansion of det(A+B+C) is given. The robustness problem for discrete-time state-space systems with uncertain parameters is investigated by applying the Schur stability theory of interval polynomials. A stability robustness criterion for the uncertain systems is derived. Illustrative examples are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

State-Space Approach to LQG Multivariable Predictive and Feedforward Optimal Control

Two extensions of the LQG optimal control problem are presented which enable prediction and feedforward action to be introduced in a state equation framework. Predictive control is first considered using a discrete-time state-space system model. Future reference values are assumed to be known p-steps ahead and the controller has a two-degrees of freedom structure. The mechanism for introducing feedforward control and hence providing a third degree of freedom is also considered. Both prediction and feedforward action are simple to design and implement using the proposed state-space approach.