The role of transmission zeros in linear multivariable regulators

The problem is considered of designing compensators to regulate linear multivariable systems. It is shown that the essential mechanism employed by such a compensator is the multivariable analogue of pole-zero cancellation: the compensator supplies transmission zeros to cancel the unstable poles of the disturbance and reference signals. If the compensator is additionally required to function in the presence of small variations in system parameters then it must supply transmission zeros in greater multiplicity. Finally it is shown that the transmission zeros are generated by an ‘ internal model ’, incorporated in the compensator, of the dynamics of the disturbance and reference signals.     

Loop shaping design related to LQG/LTR for SISO minimum phase plants

One method of model-based compensator design for linear systems consists of two stages: state feedback design and observer design. A key issue in recent work in multivariable synthesis involves selecting the observer (state feedback) gain so that the final loop transfer function is the same as the state feedback (observer) loop transfer function. This is called loop transfer recovery (LTR)(Athans and Stein 1987, Kazerooni and Houpt 1986, Kazerooni et al. 1985, Doyle and Stein 1981). This paper shows how identification of the internal mechanism of the LTR provides simple design rules with little algebra for single-input single-output (SISO) systems. In the SISO case, the LQG/LTR reduces to computation of a compensator that shapes the loop transfer function by (i) cancelling the zeros of the plant with the compensator poles, and (ii) locating a new set of zeros for the compensator to shape the loop transfer function.     

Design of pole-placement compensators for multivariable systems

The paper describes a two-stage method for the design of dynamic compensators for pole placement in linear multivariable systems. In the first stage, a number of poles are assigned by means of constant output feedback. In the second stage, the assigned poles are preserved and a number of additional poles are placed using dynamic output feedback. The pole placement is achieved using a compensator of lower order than required by the existing methods, since now the compensator has non-unity rank transfer function matrix. In particular, for a compensator of given order, a larger number of poles can be placed using this method compared to the existing methods.     

A variable time delay compensator for multivariable linear processes

A variable time delay compensator based on the Smith predictor is proposed. Finite difference and cinematic techniques are employed for a numerical approximation of the variable delay. The resulting multivariable controller is analyzed for robustness using the structured singular value. The technique is demonstrated through two simulated systems: a single-input, single-output numerical example, and a cross-direction paper machine control problem.     

Global stability of adaptive pole placement algorithms

This paper presents direct and indirect adaptive control schemes for assigning the closed-loop poles of a single-input, single-output system in both the continuous- and discrete-time cases. The resulting closed-loop system is shown to be globally stable when driven by an external reference signal consisting of a sum of sinusoids. In particular, persistent excitation of the potentially unbounded closed-loop input-output data, and hence convergence of a sequential least-squares identification algorithm is proved. The results are applicable to standard sequential least squares, and least squares with covariance reset.     

Pole assignment using dynamic output feedback compensators

The design of dynamic output feedback compensators for the pole assignment of linear time-invariant multivariable systems is considered. A systematic procedure is developed for the synthesis of the transfer function matrix of the compensator. A certain number of poles can be arbitrarily assigned by the proposed compensator. Moreover, in order to assign more poles using a compensator of the same order as before, a new approximate assignment approach based on the least-square-error algorithm is proposed. A numerical example is given to demonstrate the effectiveness of the proposed approaches.     

Adaptive exponential stabilization for a class of nonlinear retarded processes

This paper considers the problem of adaptive exponential stabilization for a class of single-input single-output nonlinear retarded processes. The class includes certain linear retarded systems which are subject to sector-bounded actuator and sensor nonlinearities. It is shown that there is a wide range of high-gain adaptive compensators which achieve exponential stability for the class of processes under consideration. This work was supported by SERC under Grant No. GR/D/45710.     

Canonical forms for linear multivariable systems

In this paper a class of well-known canonical forms for single-input or single-output controllable and observable systems are extended to multivariable systems. It is shown that, unlike the single-variable case, the canonical forms are generally not unique, but that the structure of the canonical form can be controlled to some extent by the designer. A major result of the paper is that a multi-input system can be transformed to a set of coupled single-input subsystems.     

Self-tuning regulators for a class of multivariable systems

Control of a class of multivariable systems described by linear vector difference equations with constant but unknown parameters is discussed. A multivariable minimum variance strategy is first presented. This gives a generalization of the minimum variance strategy for single-input single-output systems. A multivariable self-tuning regulator based on the minimum variance strategy is then proposed. It uses a recursive least squares estimator and a linear controller obtained directly from the current estimates. The asymptotic properties of the algorithm are discussed. If the estimated parameters converge, the resulting controller will under certain conditions give the minimum variance strategy. The analysis also gives insight into the case when several single-input single-output self-tuning regulators are operating in cascade mode.     

Computation of invariant zeros of linear,time-invariant,multivariable systems

In this paper a numerical method is presented for computing the invariant zeros of a controllable linear, time-invariant, multivariable system described by the 4-tuplo (A, B, C, D) or the triple (A, B, C). The method is based on the fact. that a controllable system can be made maximally unobservable by means of state variable feedback, thereby causing the cancellation of the invariant zeros by an equal number of the system poles. The invariant zeros are obtained as the eigenvalues of a matrix of the same dimension as the number of invariant zeros. The method is applicable to both multivariable as well as single-input, single-output systems. Examples are given to illustrate the use of the method.     

Simultaneous partial pole placement: A new approach to multimode system design

Simultaneous partial pole placement of a family of single-input single-output plants is proposed as a generalization of the classical pole placement and stabilization problems. This problem finds application in the design of a compensator for a family of linear dynamical systems. In this note we show that the proposed problem is equivalent to a new class of transcendental problem using stable, minimum phase rational functions with real coefficients. A necessary condition for the solvability of the associated transcendental problem is obtained. Finally, a counterexample to the following conjecture is obtained-"pairs of simultaneously stabilizable plants of bounded McMillan degree have simultaneously stabilizing compensators of bounded McMillan degree."     

The output control of linear time-invariant multivariable systems with unmeasurable arbitrary disturbances

Necessary and sufficient conditions are derived for a minimal order linear time-invariant differential feedback control system to exist for a linear time-invariant multivariable system with unmeasurable arbitrary disturbances of a given class occurring in it, such that the outputs of the system asymptotically become equal to preassigned functions of a given class of outputs, independent of the disturbances occurring in the system, and such that the closed-loop system is controllable. The feedback gains of the control system are obtained so that the dynamic behavior of the closed-loop system is specified by using either an integral quadratic optimal control approach or a pole assignment approach. The result may be interpreted as being a generalization of the single-input, single-output servomechanism problem to multivariable systems or as being a solution to the asymptotic decoupling problem.     

Solvability of static output-feedback stabilization for LTI positive systems

This paper addresses the stabilization problem of positive linear systems which have nonnegative states whenever the initial conditions are nonnegative. The synthesis of static output-feedback controllers that ensure the positivity and asymptotic stability of the closed-loop system is investigated. It is shown that this important problem is completely solved for single-input and single-output positive systems. The proposed approach can be applied to multi-input positive systems with controllers having one rank gains. All the provided conditions are necessary and sufficient and can be solved in terms of Linear Programming.     

Properties of optimal weighted sensitivity designs

Weighted sensitivity designs for a class of single-input/single-output, linear time-invariant systems are shown to have the property that the optimal controller has fewer right-half plane poles than the plant has right-half plane zeros. While weighted complementary sensitivity designs are shown to have the property that the optimal controller has fewer right-half plane zeros than the plant has right-half plane poles. Effects of various choices of weighting functions on the optimal solutions are described     

Continued fraction methods for the reduction of constant-linear multivariable systems

Methods developed for the reduction in order of single-input/single-output systems are extended to the reduction of linear, constant, equal input/equal output multivariable systems. The methods considered are the continued fraction and the Padé approximation techniques. Examples are given to illustrate the method.     

2-D exact model matching

The exact model matching problem in single-input single-output 2-D systems is solved by applying polyaomial techniques. All solutions are described in a parametric form. Natural solvability conditions for causal and/or stable solutions are given and a simple design procedure is proposed. It consists in solving a linear 2-D polynomial equation which directly yields a suitable compensator.     

An algebraic approach to iterative learning control

In this paper discrete-time iterative learning control (ILC) systems are analysed from an algebraic point of view. The algebraic analysis shows that a linear-time invariant single-input–single-output model can always represented equivalently as a static multivariable plant due to the finiteness of the time-axis. Furthermore, in this framework the ILC synthesis problem becomes a tracking problem of a multi-channel step-function. The internal model principle states that for asymptotic tracking (i.e. convergent learning) it is required that an ILC algorithm has to contain an integrator along the iteration axis, but at the same time the resulting closed-loop system should be stable. The question of stability can then be answered by analysing the closed-loop poles along the iteration axis using standard results from multivariable polynomial systems theory. This convergence theory suggests that time-varying ILC control laws should be typically used instead of time-invariant control laws in order to guarantee good transient tracking behaviour. Based on this suggestion a new adaptive ILC algorithm is derived, which results in monotonic convergence for an arbitrary linear discrete-time plant. This adaptive algorithm also has important implications in terms of future research work—as a concrete example it demonstrates that ILC algorithms containing adaptive and time-varying components can result in enhanced convergence properties when compared to fixed parameter ILC algorithms. Hence it can be expected that further research on adaptive learning mechanisms will provide a new useful source of high-performance ILC algorithms.     

Globally stable adaptive system design for minimum phase SISO plants with input saturation

Model reference adaptive control problem for single-input single-output time-invariant continuous-time plants with input saturation is considered with main attention focused on global properties. A sufficient condition is presented and a new design method of adaptive control systems is proposed. If a priori information about the plant is available to choose the reference model and the reference input so that the sufficient condition holds, the closed-loop adaptive control system designed by the proposed method can have global stability and globally output tracking property. It is shown that the sufficient condition becomes necessary in some cases.     

Sensitivity integral relations and design trade-offs in linearmultivariable feedback systems

The purpose of this paper is to develop integral relations regarding the singular values of the sensitivity function in linear multivariable feedback systems. The main utility of these integrals is that they can be used to quantify the fundamental limitations in feedback design which arise due to system characteristics such as open-loop unstable poles and nonminimum phase zeros and to such fundamental design requirements as stability and bandwidth constraints. We present extensions to both the classical Bode sensitivity integral relation and Poisson integral formula. These extended integral relations exhibit important insights toward trade-offs that must be performed between sensitivity reduction and sensitivity increase due to the aforementioned system characteristics and design constraints. Most importantly, these results display new phenomena concerning design limitations in multivariable systems which have no analog in single-input single-output systems     

Time-scale structure assignment in linear multivariable systems using high-gain feedback

A systematic design method for assigning a required time-scale structure to a given multivariable system is developed. The time-scale structure assignment is carried out using high-gain feedback of either the state or the output. The method developed decomposes a given multivariable system into several single-input single-output subsystems, each of which is designed separately. A standard method of design for single-input single-output systems is developed, and this forms a building block for the multivariable system design. The state feedback design depends only on the stage sequence of the given system, while the output feedback design depends only on the set of integers that specifies the order of infinite zeros. In this way, our design is made robust with respect to certain parameter uncertainties. High-gain parameter(s) can be adjusted on line, and this provides a tool for the fine tuning of the desired performance characteristics of a control system.