Linear Time-invariant Multivariable Systems Research Articles

Stability margins for multivariable linear time-invariant systems

In this paper, Singular Perturbation Margin (SPM) and Generalized Gain Margin (GGM) are used as the classical Phase Margin (PM) and Gain Margin (GM) like stability metrics for multivariable Linear Time-Invariant (LTI) systems established from the view of the singular perturbation and the regular perturbation, respectively. The bijective relationship between the SPM and PM of a multivariable LTI system is established as corresponding lemmas and theorems, which is a generalisation of the single input single output cases. The analysis process for SPM and GGM are provided here. All the theoretical results are shown by a two-input-two-output LTI system (a simple model of a two-body spacecraft with non-collocated sensors and actuators using position feedback) as an application example. Such results are one essential step towards the big map of establishing a set of both theoretically rigorous and physically meaningful stability margin metrics for NLTV systems.

Bond graph realizations of state space linear time-invariant multivariable descriptions

Given a state space description of a linear time-invariant multivariable system, a similarity transformation is applied to get a controllability canonical form. This transformation assures a Bond Graph (BG) realization and avoids the use of active bonds. Active or signal bonds do not preserve the continuity of power flows and consequently, energy conservation properties are lost. The state matrix of the system in transformed coordinates contains an antisymmetric matrix associated with the storage field and the rest is associated with the dissipative field. Thus, in general, the dissipative field is implemented with a multiport element, and in order to give physical meaning, the storage field is implemented with one-port elements. The resulting BG model does not have zero-order causal paths, that is, algebraic loops between the dissipative elements. The results are applied to the BG realization of a linear robust multivariable controller that is designed for a two mass system modelled in BG.

Optimal experiment design for multivariable system identification using simultaneous excitation

Having an accurate model of a system is essential for many applications today, especially those related to advanced process control (APC). When executing an industrial delivery project of APC, often significant time is spent performing experiments on the real process to identify a model. By designing higher quality experiments the time needed on site carrying out experiments can be reduced, saving both resources and engineering efforts. This paper investigates optimal input design to minimize the time needed to identify a linear time-invariant multi-variable system fulfilling certain requirements on the model accuracy. This is an extension of previous work which focused on sequential excitation on one input signal at the time. Here, however, the effects of using combined simultaneous and sequential excitation is investigated. The resulting non-convex optimization problem is relaxed using binary decision variables. An important feature of the approach is that the experiment is carried out in closed-loop using a model predictive controller with zone constraints to further guarantee that the output constraints are not violated. Simulation results indicate that there are many cases where using combined simultaneous and sequential excitation outperforms the previous sequential approach.

On proper and stable stabilising controllers for a class of linear multivariable systems

In this paper, we examine the problems of existence and computation of proper and stable stabilising compensators in the feedback path of a special class of linear time-invariant multivariable systems characterised by a strictly proper transfer function matrix. For the class of systems that are ‘square’, i.e. have the same number of inputs and outputs and have no zeros in the closed right half complex plane and the class of ‘non-square’ systems that include a ‘square’ subsystem that has no zeros in the closed right half complex plane we present a sufficient condition for the existence and computation of proper stabilising (pole placing) compensators which in certain cases that depend on the transfer function matrix of the system can be chosen to be stable.

Stabilization of the Pendubot: a polynomial matrix approach

Abstract This paper concerns the stabilization problem for an underactuated robot called the Pendubot. Relying on a computational algorithm which is based on various results of the ‘polynomial matrix approach’, we propose an output-feedback-based internally stabilizing controller to stabilize the Pendubot at the unstable vertical upright position. The algorithm utilizes results for the solution of polynomial matrix Diophantine equations required for the computation and parameterization of proper ‘denominator assigning’ and internally stabilizing controllers for linear time invariant multivariable systems and reduces the problem to that of the solution of a set of numerical linear equations. The controller presented uses only the measured output which consists of the angles of the two links and does not require knowledge of the angular velocities which are usually not directly measurable. Comparative simulations are carried out to verify the good performance of the proposed controller. Finally, experimental results are provided to demonstrate the validity and feasibility of the proposed method.

Stabilization of multivariable linear time-invariant systems by proportional-plus-derivative state feedback

Abstract This paper is devoted to the study of the problem of stabilization by proportional-plus-derivative state feedback for multivariable linear time-invariant systems. In particular, explicit necessary and sufficient conditions are established for the stability of a closed-loop system obtained by proportional-plus-derivative state feedback from the given multivariable linear time-invariant system. A procedure is given for the computation of stabilizing proportional-plus-derivative state feedback. Our approach is based on properties of real and polynomial matrices.

Invariant output feedback stabilisability: the scalar case

In this paper we prove that stabilisability is a static output feedback (SOF) invariant, for scalar and multivariable systems. Then we examine scalar stabilisability, from an invariant viewpoint. We prove that the signature of Hermite’s Bezoutian is constant within certain intervals that we call critical, and we give a very simple algebraic stabilisability criterion, consisted of a finite number of stability checks, one for each critical interval. We establish the validity of this criterion with Routh, Hurwitz and Lyapunov methods. We prove that the winding number of the Nyquist plot around points of the real axis, is constant within critical intervals. We correlate the stabilisability within critical intervals with the stability of corresponding critical polynomials defined at their limits. Finally, we discuss why our findings constitute a decisive step towards the analytical solution for the persisting problem of static output feedback stabilizability of linear time invariant multivariable systems.

Pole Assignment by Proportional-plus-derivative State Feedback for Multivariable Linear Time-invariant Systems

In this paper the pole assignment problem by proportional-plus-derivative state feedback for multivariable linear-time invariant systems is studied. In particular, explicit necessary and sufficient conditions are established for a given polynomial with real coefficients to be characteristic polynomial of closed-loop system obtained by proportional-plus-derivative state feedback from the given multivariable linear time-invariant system. A procedure is given for the calculation of proportional-plus-derivative state feedback which places the poles of closed-loop system at desired locations. Our approach is based on properties of polynomial matrices.

Phases of discrete-time LTI multivariable systems

In contrast to the well-developed gain analysis for multi-input-multi-output (MIMO) linear time-invariant (LTI) systems, the research on the phase analysis does not share the same status. In this paper, we introduce the phase response of a class of discrete-time (DT) LTI multivariable systems by exploiting a definition of matrix phases based on the numerical range. Positive frequency joint sectorial systems are also defined, which can generalize the positive real and negative imaginary systems. The interpretation of system phases is given in terms of the input and output signals of the system. A sectored real lemma is obtained to characterize the phase information from a state–space realization. Motivated by finding a phasic counterpart to the small gain theorem, we develop a small phase theorem for the internal stability of a closed-loop system. The phase properties are also investigated for parallel and feedback interconnected systems.

Unknown input observer design for linear time-invariant multivariable systems based on a new observer normal form

In various applications in the field of control engineering, the estimation of the state variables of dynamic systems in the presence of unknown inputs plays an important role. Existing methods require the so-called observer matching condition to be satisfied, rely on the boundedness e variables or exhibit an increased observer order of at least twice the plant order. In this article, a novel observer normal form for strongly observable linear time-invariant multivariable systems is proposed. In contrast to classical normal forms, the proposed approach also takes the unknown inputs into account. The proposed observer normal form allows for the straightforward construction of a higher-order sliding mode observer, which ensures global convergence of the estimation error within finite time even in the presence of unknown bounded inputs. Its application is not restricted to systems which satisfy the aforementioned limitations of already existing unknown input observers. The proposed approach can be exploited for the reconstruction of unknown inputs with bounded derivative and robust state-feedback control, which is shown by means of a tutorial example. Numerical simulations confirm the effectiveness of the presented work.

Linear-Convex Optimal Steady-State Control

We consider the problem of designing a feedback controller for a multivariable linear time-invariant system, which regulates a system output to the solution of an equality-constrained convex optimization problem despite unknown constant exogenous disturbances; we term this the linear-convex optimal steady-state (OSS) control problem. We introduce the notion of an optimality model, and show that the existence of an optimality model is sufficient to reduce the OSS control problem to a stabilization problem. This yields a design framework for OSS control that unifies and extends existing design methods in the literature. We illustrate the approach via an application to optimal frequency control of power networks, where our methodology recovers centralized and distributed controllers reported in the recent literature.

Phase Analysis for Discrete-time LTI Multivariable Systems

In contrast to the well-developed gain analysis for multi-input-multi-output (MIMO) linear time-invariant (LTI) systems, the research on the phase analysis does not share the same status. In this paper, we introduce the phase response of a class of discrete-time (DT) LTI multivariable system by exploiting a definition of matrix phases based on the numerical range. Half-sectorial transfer matrices are defined, which can generalize the positive real and negative imaginary systems. A sectored real lemma is obtained to characterize the phase information of a half-sectorial system from a state-space realization. Motivated by finding a phasic counterpart to the small gain theorem, we develop a small phase theorem for the internal stability of closed-loop systems.

Adaptive control of linear multivariable systems using dynamic regressor extension and mixing estimators: Removing the high-frequency gain assumptions

In this paper it is shown that, using the recently introduced dynamic regressor extension and mixing parameter estimation technique, it is possible to remove the key assumption of prior knowledge on the high frequency gain imposed in model reference adaptive control of linear time-invariant multivariable systems. For the case of scalar systems this is tantamount to removing the requirement of known sign of this coefficient. The proposed controller is not certainty equivalent in the sense that the parameters generated by the new estimator are not directly applied to the controller, but they are modified adding a shifting and a scaling factor. Under the weak assumption of interval excitation, parameter convergence in finite time of the modified parameters is achieved, ensuring good robustness of the proposed controller.

Reduced-order multiple observer for Takagi–Sugeno systems with unknown inputs

The paper discusses a niche area problem — the design of reduced-order multiple observers which can achieve the finite-time state reconstruction for Takagi–Sugeno multiple models with unknown inputs. The new reduced-order multiple observer obtained in this paper is only the second ever designed, the author of this paper continuing his work started one year ago with the introduction of the reduced-order multiple observer concept. The obtaining of the new reduced-order multiple observer is achieved by combining two classical reduced-order observers for linear time-invariant multivariable systems with unknown inputs and a full-order multiple observer for Takagi–Sugeno systems. The main innovative idea behind the design of the reduced-order multiple observer for Takagi–Sugeno systems described by the multiple models is the split of the multiple model into two subsystems: an unknown-input-free subsystem and an unknown-input-depending subsystem. The unknown-input-free subsystem is brought to the form of a standard multiple model and, in this situation, any algorithm to design full-order multiple observers can be used in the design of reduced-order multiple observers. The steps of the design procedure have been summarized and software implemented; then, the validation of the suggested algorithm has been done for two concrete cases associated to the motion of an aircraft during its landing

Iterative algorithms for the input and state recovery from the approximate inverse of strictly proper multivariable systems

Abstract This paper proposes new iterative algorithms for the unknown input and state recovery from the system outputs using an approximate inverse of the strictly proper linear time-invariant (LTI) multivariable system. One of the unique advantages from previous system inverse algorithms is that the output differentiation is not required. The approximate system inverse is stable due to the systematic optimal design of a dummy feedthrough D matrix in the state-space model via the feedback stabilization. The optimal design procedure avoids trial and error to identify such a D matrix which saves tremendous amount of efforts. From the derived and proved convergence criteria, such an optimal D matrix also guarantees the convergence of algorithms. Illustrative examples show significant improvement of the reference input signal tracking by the algorithms and optimal D design over non-iterative counterparts on controllable or stabilizable LTI systems, respectively. Case studies of two Boeing-767 aircraft aerodynamic models further demonstrate the capability of the proposed methods.

Pole assignment of multivariable systems using proportional-derivative state feedback

ABSTRACTIn this article, the parametric solution to the pole assignment problem for multivariable linear time-invariant systems controlled by proportional-derivative (PD) state feedback is developed. The new expressions for the PD gain controllers are derived which describe the available degrees of freedom offered by PD state feedback. The freedom provided by PD state gain matrices is utilised to obtain closed-loop systems with robust and small gain matrices. Two computational algorithms are introduced, and their effectiveness is demonstrated by two simulation examples.

Matchable-Observable Linear Models and Direct Filter Tuning: An Approach to Multivariable Identification

Identification of linear time-invariant multivariable systems can best be understood as comprising three separate problems: selection of system model structure, filter design, and parameter estimation itself. Approaching the first using matchable-observable models originally developed in the adaptive control literature and the second via direct or derivative-free optimization, effective least-squares algorithms can be used for parameter estimation. The accuracy, robustness and moderate computational demands of the methods proposed are demonstrated via simulations with randomly generated models and applied to identification using real process data. The results obtained are comparable or superior to the best results obtained using standard implementations of the algorithms described in the literature.

Geometric structure and properties of linear time invariant multivariable systems in the controller canonical form

In this paper, we analyse some fundamental structural properties of linear time-invariant multivariable systems in the controller canonical form and present a direct method for the computation of bases and associated friends for output-nulling, input-containing and reachability subspaces in terms of the parameters of the system and the invariant zero structure, both in the nondefective and in the defective case. Using this analysis, it is possible to express the solvability conditions of important control and estimation problems in terms of easily checkable conditions on the system matrices.

Fixed-Order Decoupling of Interval Plant Families

A method for the reduction of interactions in linear time invariant (LTI) multivariable uncertain systems is proposed. An H∞-norm metric is proposed for the assessment of interactions in interval uncertain multiple-input multiple-output (MIMO) plants. Based on this, a procedure for the design of fixed-order dynamic decoupling precompensators for MIMO plants with interval uncertainty is outlined which can be solved using efficient solvers such as cvx. The proposed methodology is used to develop a low-order robust multivariable controller for voltage and frequency control of an islanded distributed generation (DG) unit.

Full Order Unknown Inputs Observer for Multiple Time-Delay Systems

Abstract In this paper, a design of a full-order observer for linear time-invariant (LTI) multivariable systems with multiple time-delays and unknown inputs (UI) is proposed. The main idea is to reduce the problem of the unknown input observer (UIO) for systems with multiple time-delays to that of a standard one. To that purpose, the orthogonal collocation method is used to transform the infinite dimensional model of the delayed system described by a set of linear partial differential equations (PDEs) to a finite dimensional one described by a set of linear ordinary differential equations (ODEs). Even using an approximation method, the asymptotic stability of the UIO is well proven. The efficiency of the proposed algorithm is shown using the quadruple-tank benchmark. The two cases of minimum and non-minimum phase models are considered.