Minimal State-space Representation Research Articles

Analysis of the implicit Euler time-discretization of passive linear descriptor complementarity systems

paper 7269 This article is largely concerned with the time-discretization of descriptor-variable systems coupled to with complementarity constraints. They are named descriptor-variable linear complementarity systems (DVLCS). More speci cally passive DVLCS with minimal state space representation are studied. The Euler implicit discretization of DVLCS is analysed: the one-step non-smooth problem (OSNSP), that is a generalized equation, is shown to be well-posed under some conditions. Then the convergence of the discretized solutions is studied. Several examples illustrate the applicability and the limitations of the developments.

Minimal State-Space Representation of Convolutional Product Codes

In this paper, we study product convolutional codes described by state-space representations. In particular, we investigate how to derive state-space representations of the product code from the horizontal and vertical convolutional codes. We present a systematic procedure to build such representation with minimal dimension, i.e., reachable and observable.

Unimodular equivalence and similarity for linear systems

ABSTRACTThe problem of finding the mapping between unimoduar transformations relating two minimal matrix fraction descriptions (MFDs) of a transfer function, and the similarity transformations relating the respective minimal state-space representations is considered. It is shown that the problem is equivalent to finding the relation of MFDs of the input-state transfer functions of the two systems. This relation turns out to be an equivalence relation involving the unimodular and the similarity matrices relating the MFDs and the state-space systems, respectively. A canonical form for MFDs under this equivalence relation is obtained and it is shown that it leads to a canonical state-space representation, via a realisation procedure.

Multi‐input and multi‐output proportional‐integral‐derivative controller design via linear quadratic regulator‐linear matrix inequality approach

This study considers the problem of designing a multi-input and multi-output (MIMO) proportional-integral-derivative (PID) controller via direct optimal or suboptimal linear quadratic regulator (LQR) approach. To design the controller, first the MIMO PID design problem is transformed into a state feedback control and then the gains of the state feedback controller are chosen through an optimal or suboptimal LQR design. Given a minimal state space representation (A, B, C) of the plant, a necessary and sufficient condition (based on matrices A, C) for which the optimal problem (i.e. PID design via optimal LQR) is solvable is obtained. When this optimal problem is not solvable, a suboptimal solution (i.e. PID design via suboptimal LQR), if exists, is obtained by converting the problem into trace minimisation one, which is solved using linear matrix inequality-based method. Suitable examples are considered to illustrate the approaches.

Global Identification of Linearized DSGE Models

This paper introduces a time domain framework to analyze global identification of stochastically nonsingular DSGE models. A formal identification condition is established that relies on the restrictions linking the observationally equivalent minimal state space representations and on the inherent constraints imposed by them on deep model parameters. We next develop an algorithm that checks global identification by searching for observationally equivalent model parametrizations. The algorithm is efficient as the identification conditions it employs shrink considerably the space of candidate deep parameter points and does not require solving the model at each of these points. We also derive two complementary necessary conditions for global identification. Their usefulness and the working of the algorithm are illustrated with an example.

On Stability in the Presence of Analog Erasure Channel Between the Controller and the Actuator

Consider a discrete-time networked control system, in which the controller has direct access to noisy measurements of the output of the plant. However, information flows from the controller to the actuator via a channel that features Bernoulli erasure events. If an erasure occurs, the channel outputs an erasure symbol; otherwise, it transmits a real finite-dimensional vector. We determine necessary and sufficient conditions for the stabilizability of an unstable linear time-invariant finite-dimensional plant. Given a minimal state-space representation for the plant, the necessary and sufficient conditions for stabilizability are expressed in terms of the probability of erasure at the channel and of the spectral radius of the one-step state transition matrix. There are two main results in the technical note. The first result shows that if the actuator has processing capabilities, then the necessary and sufficient conditions for stabilizability remain unchanged with or without acknowledgements from the actuator to the controller. The second result shows that the stabilizability conditions are identical for two types of actuators: (Type I) Processing at the actuator has access to the plant's model; (Type II) Processing at the actuator uses a universal algorithm that does not depend on the model of the plant. Thus, neither the knowledge of the model of the plant at the actuator, nor the presence of acknowledgements from the actuator to the controller, can be used to alter or relax the necessary and sufficient conditions for stabilizability.

On Bumps and Reduction of Switching Transients in Multicontroller Systems

This paper is concerned with the realization and implementation of multicontroller systems, consisting of several linear controllers, subject to the bump phenomenon which occurs when switching between one controller acting in closed loop and another controller in the set of “offline” controllers waiting to take over the control loop. Based on a deep characterization of the bump phenomenon, the paper gives a novel and simple parameterization of such set of linear controllers, possibly having different state dimensions, to cope with bumps and their undesirable transients in switched-mode systems. The proposed technique is based on a non minimal state-space representation allowing a common memory and a unique dynamics shared by all controllers in that set. It also makes each initially open-loop unstable controller run in a stable way regardless of whether that controller is connected to the controlled process.

A fully automated recurrent neural network for unknown dynamic system identification and control

This paper presents a fully automated recurrent neural network (FARNN) that is capable of self-structuring its network in a minimal representation with satisfactory performance for unknown dynamic system identification and control. A novel recurrent network, consisting of a fully-connected single-layer neural network and a feedback interconnected dynamic network, was developed to describe an unknown dynamic system as a state-space representation. Next, a fully automated construction algorithm was devised to construct a minimal state-space representation with the essential dynamics captured from the input-output measurements of the unknown system. The construction algorithm integrates the methods of minimal model determination, parameter initialization and performance optimization into a systematic framework that totally exempt trial-and-error processes on the selections of network sizes and parameters. Computer simulations on benchmark examples of unknown nonlinear dynamic system identification and control have successfully validated the effectiveness of the proposed FARNN in constructing a parsimonious network with superior performance

Minimal representations of continuous-time processes having spectral density with zeros in the extended imaginary axis

Given a process y with stationary increments and rational incremental spectral density Φ ( s ) , we parametrize minimal state-space representations of d y . We consider the case when Φ ( s ) may have zeros on the imaginary axis including infinity and we analyze the zero structure of each state-space representation.

On the State-Space Representation of Local Affine Models in Takagi–Sugeno–Kang Fuzzy Systems (Zur Zustandsdarstellung von Takagi–Sugeno–Kang-Fuzzy-Systemen mit Lokal Affinen Modellen)

Abstract Considering the constant terms in the local models of dynamic Takagi–Sugeno–Kang fuzzy models can provide for improved prediction quality. However, the constant terms have typically not been treated analytically in analysis and controller design. In this paper, minimal state-space representations are derived from local affine input/output models for usage in TSK fuzzy models. Transformation algorithms for structure, parameters and initial values are proposed.

The input-output and controllable behavioural structures of a state space representation

Given the matrices of a linear state space representation, we find an expression for a universal left annihilator of the matrix zI - A -C and hence derive kernel representations for the input-output behaviour and by duality the controllable part. More generally in the discrete case we derive representations for the L -completion for different values of L, and the subbehaviours of trajectories reachable in a given time interval. The representations are in certain trim canonical forms, which are intimately connected with structure indices. As a by-product of the state elimination procedure, we obtain a minimal state space representation for a given behaviour in terms of an arbitrary one.

MIMO systems-transfer function to state-space

The authors present a very simple method, useful for classroom teaching, to obtain a minimal state-space representation (with the exception of systems where there is a pole-zero cancellation) from a transfer function matrix. The method is direct and does not involve the intermediate step of obtaining nonminimal realization-which further requires system reduction routines-and hence is suitable as a compact self-contained topic which has been hitherto neglected in undergraduate linear control theory curriculum.

Robust Analysis of Distillation Column Dynamics

The robust stability and performance of a binary distillation column model is analysed using a minimal state space representation. The uncertainty modelling of the column is resulted in an LFT form with a four element diagonal block structure taking into account the dependence between the steady state gains. Robust analysis of the distillation column dynamics is performed and compared using µ- and l1 -analysis tools. The magnitude of possible overshoots is determined by l1-test for robust performance and the dependence of this magnitude on the dominant time constant (i.e. the purity of the separation) is investigated. Futhermore the effect of the mi certainties in the time constants and the singular values on the robust performance is also analysed using l1-test. It has been found that the uncertainty in the time constants affects dominantly the robust performance of the open-loop system, i.e. the magnitude of possible overshoots.

On coprime factorization and minimal realization of transfer function matrices using the pseudo-observability concept

A simple computational algorithm is presented for calculating the coprime matrix fraction description and minimal state-space representation of a multivariable linear system specified by a transfer function matrix. The concept of admissibility of pseudo-observability indices is utilized in the algorithm to permit extraction of the most convenient representations, in the sense that the representation elements have the lowest absolute values. A computational example is given to show the feasibility of the suggested method

Robust Control of Flexible Structures Using Multiple Shape Memory Alloy Actuators

The design and implementation of control strategies for large, flexible smart struc tures presents challenging problems. To demonstrate the capabilities of shape-memory-alloy (SMA) actuators, we have designed and fabricated a three-mass test article with multiple shape-memory- alloy, NiTiNOL, actuators. Both force and moment actuators were implemented on the structure to examine the effects of control structure interaction and to increase actuation force. These SMA actu ators exhibit nonlinear effects due to dead band and saturation. The first step in the modeling process was the experimental determination of the transfer function matrix derived from frequency response data. A minimal state space representation was determined based on this transfer function matrix. Finally, a reduced order state space model was derived from the minimal state space representation. The simplified analytical models were compared with models developed by structural identification techniques based on vibration test data. From the reduced order model, a controller was designed to dampen vibrations in the test bed. To minimize the effects of uncertainties on the closed-loop system performance of smart structures, a linear quadratic Gaussian loop transfer recovery, LQG / LTR, control methodology was utilized. A standard LQG/LTR controller was designed; however, this controller could not achieve the desired performance robustness due to saturation effects. Therefore, a modified LQG / LTR design methodology was implemented to accommodate for the limited control force provided by the actua tors. The closed-loop system response of the multiple input multiple output (MIMO) test article with robustness verification was experimentally obtained and is presented in this paper. The modified LQG / LTR controller demonstrated performance and stability robustness to both sensor noise and parameter variations.

MFDs of spinning satellite and attitude control using gyrotorquers

Questions related to the matrix fraction descriptions (MFDs) and the minimal realizations of the transfer matrix of a spinning satellite system and control system design using output feedback are considered. The control torques are generated using gyrotorquers, and the output variables are the attitude angles. Two minimal state-space representations of the system, namely, a controller-form and an observer-form realization, are obtained from a right and left MFD of the transfer matrix, respectively. Using these canonical realizations, analytical expression for the feedback matrices of the controller and the observer as functions of the system parameters are obtained. The poles of the closed-loop system are invariant with respect to the spin rate of the satellite. Simulation results are presented to show that precise attitude control is accomplished in the closed-loop system using output feedback. >

On the internal structure of minimal stochastic realizations

The internal structure of minimal state-space representations of stationary vector processes is investigated. A necessary and sufficient condition for order minimality is given along with an upper bound on the minimal order. The correspondence between the all-pass structure of a multivariable system and the minimality of the output representation is also specified.

On the minimal state-space realizations of all-pole and all-zero 2-D systems

A simple method is presented for the minimal state-space representation, described by the Roesser model, of a 2-D all-pole and all-zero transfer function. The state-space representation is derived, from a block diagram, by inspection. General expressions for the system matrices (A, b, c) of very simple form are derived explicitely in terms of the parameters of the transfer function.