The dual algebraic Riccati equations and the set of all solutions of the discrete-time Riccati equation

In this paper, two new pairs of dual continuous-time algebraic Riccati equations (CAREs) and dual discrete-time algebraic Riccati equations (DAREs) are proposed. The dual DAREs are first studied with some nonsingularity assumptions on the system matrix and the parameter matrix. Then, in the case of singular matrices, a generalised inverse is introduced to deal with the dual DARE problem. These dual AREs can easily lead us to an iterative procedure for finding the anti-stabilising solutions, especially to DARE, by means of that for the stabilising solutions. Furthermore, we provide the counterpart results on the set of all solutions to DARE inspired by the results for CARE. Two examples are presented to illustrate the theoretical results.     

Derivation of the maximum entropy H ∞-controller and a state-space formula for its entropy

The problem of maximizing the entropy of a stabilized closed-loop system is solved. The solution exploits the parametrization of all closed-loop systems that meet an H _∞-norm bound. The ‘central solution’ of this set is shown to maximize the entropy at infinity and for this case a particularly simple formula is derived for the maximum value of the entropy.     

Strong solutions of the discrete-time algebraic Riccati equation

Conditions are given under which a solution of the DARE is positive semidefinite if and only if all the eigenvalues of its associated closed-loop matrix are in the closed unit disc.     

The set of positive semidefinite solutions of the algebraic Riccatiequation of discrete-time optimal control

In this paper the algebraic Riccati equation (ARE) of the discrete-time linear-quadratic (LQ) optimal control problem and its set of positive semidefinite solutions is studied under the most general assumption which is output stabilizability. With respect to an appropriate basis, the discrete-time algebraic Riccati equation (DARE) decomposes into a Lyapunov equation and an irreducible Riccati equation. The focus is on the Riccati part which amounts to studying a DARE where all unimodular modes are controllable. A bijection between positive semidefinite solutions and certain well-defined sets of F-invariant subspaces is established which, together with its inverse, is order reversing. As an application, issues concerning positive definite or strong solutions are clarified. Analogous results for negative semidefinite solutions are valid only under an additional assumption on the unobservable subspace     

Convergence properties of constrained linear system under MPC control law using affine disturbance feedback

This paper shows new convergence properties of constrained linear discrete time system with bounded disturbances under Model Predictive Control (MPC) law. The MPC control law is obtained using an affine disturbance feedback parametrization with an additional linear state feedback term. This parametrization has the same representative ability as some recent disturbance feedback parametrization, but its choice together with an appropriate cost function results in a different closed-loop convergence property. More exactly, the state of the closed-loop system converges to a minimal invariant set with probability one. Deterministic convergence to the same minimal invariant set is also possible if a less intuitive cost function is used. Numerical experiments are provided that validate the results.     

Controller parametrization for time-varying multirate plants

The authors obtain a parametrization of all stabilizing controllers for a given discrete-time time-varying linear finite-dimensional multirate system. The parametrization is a natural extension of the Youla parametrization of all stabilizing controllers to the setting of multirate systems. The set of all stabilizing controllers is shown to be the set of all closed-loop systems obtained by terminating a fixed two-port multirate system in a stable multirate system. State-space formulas for the parametrization are also given     

Predictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions

This article presents a novel model predictive control (MPC) scheme that achieves input-to-state stabilization of constrained discontinuous nonlinear and hybrid systems. Input-to-state stability (ISS) is guaranteed when an optimal solution of the MPC optimization problem is attained. Special attention is paid to the effect that sub-optimal solutions have on ISS of the closed-loop system. This issue is of interest as firstly, the infimum of MPC optimization problems does not have to be attained and secondly, numerical solvers usually provide only sub-optimal solutions. An explicit relation is established between the deviation of the predictive control law from the optimum and the resulting deterioration of the ISS property of the closed-loop system. By imposing stronger conditions on the sub-optimal solutions, ISS can even be attained in this case.     

Frequency response estimation for normalized coprime factors

This paper investigates estimating the frequency response of the normalized coprime factors of a possibly unstable transfer function from closed-loop frequency domain experimental data. A stochastic framework has been adopted. The dual Youla-Kucera parametrization approach in closed-loop system identification has been developed to guarantee the normality of the estimated frequency response. A parametrization of all the normalized coprime factors has been established for transfer functions internally stabilizable by a known controller. Based on this parametrization, interpolation conditions have been derived for the measurement errors being consistent with the stabilizing controller and experimental data. An analytic expression has been obtained for the maximum likelihood estimate of the frequency response. The effectiveness of the suggested method is confirmed through numerical simulations.     

Constrained linear system with disturbance: Convergence under disturbance feedback

This paper proposes a disturbance-based control parametrization under the Model Predictive Control framework for constrained linear discrete time systems with bounded additive disturbances. The proposed approach has the same feasible domain as that obtained from parametrization over the family of time-varying state feedback policies. In addition, the closed-loop system is stable in the sense that the state converges to a bounded set that has a characterization determined by a feedback gain.     

Closed-loop identification of multivariable systems: With or without excitation of all references?

The accuracy of plant parameters estimated in closed-loop operation is investigated for a class of multivariable systems and for the situation where only some of the reference inputs are excited. This issue is important for at least two reasons: (i) there are control applications where it is preferable not to excite all references in order to avoid performance degradation, and (ii) it is not clear whether the results in the context of open-loop identification, where the existence of common parameters between the various transfer functions is a condition for improved accuracy with additional excitation, also hold in the closed-loop case. The paper examines the effect of the non-excitation of some reference inputs on the variance of the estimated parameters. The proposed expressions are valid for all conventional model structures used in prediction error identification. Although exciting all reference inputs is not necessary for identifiability, this work shows that, regardless of the parametrization, the excitation of all references never worsens and, in most cases, improves the accuracy of the parameter estimates. The analytical results developed in this work are illustrated by two simulation examples.     

Lower bounds for the trace of the solution of the discretealgebraic Riccati equation

Two lower bounds for the trace of the solution of the discrete algebraic Riccati equation (DARE) are presented. It is shown that in many cases, these trace bounds are tighter than those in the literature and greater than the trace of the state weighting matrix even when the system matrix is singular. The results are illustrated by an example     

Rational general solutions of planar rational systems of autonomous ODEs

In this paper, we provide an algorithm to compute explicit rational solutions of a rational system of autonomous ordinary differential equations (ODEs) from its rational invariant algebraic curves. The method is based on the proper rational parametrization of these curves and the fact that by linear reparametrizations, we can find the rational solutions of the given system of ODEs. Moreover, if the system has a rational first integral, we can decide whether it has a rational general solution and compute it in the affirmative case.     

REMI: A framework of reusable elements for mining heterogeneous data with missing information

Applications targeting smart cities tackle common challenges, however solutions are seldom portable from one city to another due to the heterogeneity of smart city ecosystems. A major obstacle involves the differences in the levels of available information. In this work, we present REMI, which is a mining framework that handles varying degrees of information availability by providing a meta-solution to missing data. The framework core concept is the REMI layered stack architecture, offering two complementary approaches to dealing with missing information, namely data enrichment (DARE) and graceful degradation (GRADE). DARE aims at inference of missing information levels, while GRADE attempts to mine the patterns using only the existing data.We show that REMI provides multiple ways for re-usability, while being fault tolerant and enabling incremental development. One may apply the architecture to different problem instantiations within the same domain, or deploy it across various domains. Furthermore, we introduce the other three components of the REMI framework backing the layered stack. To support decision making in this framework, we show a mapping of REMI into an optimization problem (OTP) that balances the trade-off between three costs: inaccuracies in inference of missing data (DARE), errors when using less information (GRADE), and gathering of additional data. Further, we provide an experimental evaluation of REMI using real-world transportation data coming from two European smart cities, namely Dublin and Warsaw.     

Parametric state-feedback control with response insensitivity

A recent parametrization of the class of linear state-feedback controllers that assign a set of desired self-conjugate eigenvalues to the closed-loop system is used to formulate and solve a fundamental response insensitivity problem. It is established to what extent state-feedback control can be used to render the closed-loop system response insensitive to possibly many not necessarily small parameter variations in the open-loop state-space model. A non-conservative sequential design procedure is developed for making as many of the closed-loop system eigenmodes as possible totally insensitive while retaining arbitrary assignment of the maximum number of closed-loop eigenvalues. The main result is a class of desensitizing fixed-gain state-feedback controllers explicitly specified by a set of free parameters which may be chosen to satisfy additional design requirements.     

Robust maximum-likelihood estimation of multivariable dynamic systems

This paper examines the problem of estimating linear time-invariant state-space system models. In particular, it addresses the parametrization and numerical robustness concerns that arise in the multivariable case. These difficulties are well recognised in the literature, resulting (for example) in extensive study of subspace-based techniques, as well as recent interest in ‘data driven’ local co-ordinate approaches to gradient search solutions. The paper here proposes a different strategy that employs the expectation-maximisation (EM) technique. The consequence is an algorithm that is iterative, with associated likelihood values that are locally convergent to stationary points of the (Gaussian) likelihood function. Furthermore, theoretical and empirical evidence presented here establishes additional attractive properties such as numerical robustness, avoidance of difficult parametrization choices, the ability to naturally and easily estimate non-zero initial conditions, and moderate computational cost. Moreover, since the methods here are maximum-likelihood based, they have associated known and asymptotically optimal statistical properties.     

Global adaptive control of nonlinearly parametrized systems

In this paper, we consider global adaptive control of nonlinearly parametrized systems in parametric-strict-feedback form. Unlike previous results, we do not require a priori bounds on the unknown parameters, which is as in the linear parametrization case. We also allow unknown parameters to be time-varying provided they are bounded. Our proposed adaptive controller is a switching type controller, in which the controller parameter is tuned in a switching manner via a switching logic. Global stability results of the closed-loop system have been proved.     

Reduced order optimization for model predictive control using principal control moves

In order to reduce the computational complexity of model predictive control (MPC) a proper input signal parametrization is proposed in this paper which significantly reduces the number of decision variables. This parametrization can be based on either measured data from closed-loop operation or simulation data. The snapshots of representative time domain data for all manipulated variables are projected on an orthonormal basis by a Karhunen-Loeve transformation. These significant features (termed principal control moves, PCM) can be reduced utilizing an analytic criterion for performance degradation. Furthermore, a stability analysis of the proposed method is given. Considerations on the identification of the PCM are made and another criterion is given for a sufficient selection of PCM. It is shown by an example of an industrial drying process that a strong reduction in the order of the optimization is possible while retaining a high performance level.     

Adaptive friction compensation of servo mechanisms

In this paper, adaptive friction compensation is investigated using both model-based and neural network (non-model-based) parametrization techniques. After a comprehensive list of commonly used models for friction is presented, model-based and non-modelbased adaptive friction controllers are developed with guaranteed closed-loop stability. Intensive computer simulations are carried out to show the effectiveness of the proposed control techniques, and to illustrate the effects of certain system parameters on the performance of the closed-loop system. It is observed that as the friction models become complex and capture the dominate dynamic behaviours, higher feedback gains for model-based control can be used and the speed of adaptation can also be increased for better control performance. It is also found that neural networks are suitable candidate for friction modelling and adaptive controller design for friction compensation.     

Adaptive control of MIMO time-varying systems with indicator function based parametrization

This paper studies a new solution framework for adaptive control of a class of MIMO time-varying systems with indicator function based parametrization, motivated by a general discrete-time MIMO Takagi–Sugeno (T–S) fuzzy system model in an input–output form with unknown parameters. An indicator (membership) function based parametrization has some favorable capacity to deal with certain large parameter variations. A new discrete-time MIMO system prediction model is derived for approximating a nonlinear dynamic system, and its system properties are clarified. An adaptive control scheme is developed, with desired controller parametrization and stable parameter estimation for control of such uncertain MIMO time-varying systems. A control singularity problem is addressed and the closed-loop stability and output tracking properties are analyzed. This work provides a new method for multivariable T–S fuzzy system modeling and adaptive control. An illustrative example and simulation results are presented to demonstrate the proposed novel concepts and to verify the desired adaptive control system performance.     

Dirac structures on Hilbert spaces and boundary control of distributed port-Hamiltonian systems

Aim of this paper is to show how the Dirac structure properties can be exploited in the development of energy-based boundary control laws for distributed port-Hamiltonian systems. Usually, stabilization of non-zero equilibria has been achieved by looking at, or generating, a set of structural invariants, namely Casimir functions, in closed-loop. Since this approach fails when an infinite amount of energy is required at the equilibrium (dissipation obstacle), this paper illustrates a novel approach that enlarges the class of stabilizing controllers. The starting point is the parametrization of the dynamics provided by the image representation of the Dirac structure, that is able to show the effects of the boundary inputs on the state evolution. In this way, energy-balancing and control by state-modulated source methodologies are extended to the distributed parameter scenario, and a geometric interpretation of these control techniques is provided. The theoretical results are discussed with the help of a simple but illustrative example, i.e. a transmission line with an RLC load in both serial and parallel configurations. In the latter case, energy-balancing controllers are not able to stabilize non-zero equilibria because of the dissipation obstacle. The problem is solved thanks to a (boundary) state-modulated source.