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An algorithm is proposed to compute a state feedback gain matrix for a linear time invariant, regular descriptor system, which ensures that i) the closed loop system is impulse-free and ii) all the finite poles are placed within a pre-defined stability region in the complex plane. The associated design flexibilities are exploited in minimizing upper bounds of the Frobenius norm of respective gain matrices. A class of linear matrix inequality (LMI) regions in the complex plane are chosen as constraints for the closed loop finite poles. By representing a subset of the nonsingular matrices through solution of an LMI, and linearizing the set of matrix inequalities, arise in the regional pole (finite poles) assignment, the associated optimizations are formulated as semidefinite programs. The effectiveness of the developed algorithm is demonstrated through numerical examples. Significant reduction in the norm of feedback gain matrix is achieved in a two generator six bus power system.

This paper is concerned with the energy shaping of 1-D linear boundary controlled port-Hamiltonian systems. The energy-Casimir method is first proposed to deal with power preserving systems. It is shown how to use finite dimensional dynamic boundary controllers and closed-loop structural invariants to partially shape the closed-loop energy function and how such controller finally reduces to a state feedback. When dissipative port-Hamiltonian systems are considered, the Casimir functions do not exist anymore (dissipation obstacle) and the immersion (via a dynamic controller)/reduction (through invariants) method cannot be applied. The main contribution of this paper is to show how to use the same ideas and state functions to shape the closed-loop energy function of dissipative systems through direct state feedback i.e. without relying on a dynamic controller and a reduction step. In both cases, the existence of solution and the asymptotic stability (by additional damping injection) of the closed-loop system are proven. The general theory and achievable closed-loop performances are illustrated with the help of a concluding example, the boundary stabilization of a longitudinal beam vibrations.

Classical affine root locus applies to controllers that are linear in a parameter $k$ and yield affinely parameterized closed-loop denominator polynomials. This paper presents root locus rules for controllers that are rational in $k$ and yield quadratically parameterized closed-loop denominator polynomials. We show that the quadratic root locus has several advantages relative to the classical affine root locus. Specifically, the quadratic root locus is high-parameter stabilizing for minimum-phase systems that are relative degree 1, 2, or 3. Moreover, the quadratic root locus admits a wider variety of asymptote angles, which provides more controller design flexibility. The quadratic root locus is illustrated on several examples that are difficult to handle using affine root locus, such as high-parameter stabilization of the triple integrator.

Lur'e systems represent feedback interconnections of a Linear Time-Invariant system with a nonlinear operator. The Zames-Falb criterion is a powerful tool to determine the stability of a Lur'e system when the nonlinear operator is defined by a monotonic, odd and single-valued function. The article provides a generalization of the Zames-Falb criterion for the analysis of stability of a Lur'e system for nonlinear static operators that are “approximately” monotonic and odd. This result extends the standard Zames-Falb criterion with an additional notion of “robustness”.

We consider a model predictive control framework that includes a discrete-time linear time-invariant nominal plant model augmented with an output integrator disturbance model and a Kalman filter to estimate the state and disturbance vectors. While the application of this framework can guarantee offset-free control, it has shown a consistent limitation in the achievable closed loop estimator performance. Using root locus techniques, we identify sufficient conditions for a class of nominal plant models with at least one real pole for which the closed loop estimator poles cannot be arbitrarily selected regardless of the augmented system's statistics. We present several examples illustrating the limitations of the closed loop estimator pole locations.

This paper develops a root locus technique for random reference tracking in systems with saturating actuators. This is accomplished by introducing the notion of S-poles, which are the poles of the quasilinear system obtained by applying the method of stochastic linearization to the original system with saturation. The path traced by the S-poles on the complex plane when the gain of the controller changes from 0 to infin is the S-root locus. We show that the S-root locus is a subset of the standard root locus, which may terminate prematurely in the so-called termination points. A method for calculating these points is presented and a number of other, more subtle, differences between the usual and the S-root loci are described. In addition, the issue of amplitude truncation in terms of the S-root locus is investigated. Finally, an application of the S-root locus to hard disk drive controller design is presented.

The properties of the PID gain set for which all the closed-loop poles lie in the left half of the complex plane are examined on the parameter plane. A new method of determining the increase in the number of unstable poles when the gain crosses the boundary of stability regions is proposed. This method is useful for determining the number of unstable poles in each region. These results are extended to Gamma-stability based on conformal mapping.

A new phase-lead design method using the root locus diagrams is proposed. In the traditional phase-lead root locus design procedure, the designer has no direct control to the steady-state error constant of the resulting closed-loop system. In order to obtain a satisfactory steady-state error constant, the designer has to try different locations of the compensator's zero and the desired dominant roots. However, the procedure does not indicate to the designer how to alter the locations to obtain the desired steady-state error constant. With the new method presented in this note, the applicability of the phase lead compensation network is determined once the desired dominant roots are given, and the maximum "reachable" steady-state error constant can be directly determined. If the desired steady-state error constant rely within the "reachable" limit, the desired steady-state error constant together with the desired dominant roots are directly used to obtain the compensator. If the desired stead-state error constant exceeds the reachable limit, the note proposes a way to find out the proper locations of the desired dominant roots.

This paper is devoted to stability problems of automatic control systems. Frequency domain criteria for global asymptotic stability of equilibrium sets for new classes of nonlinearities are derived. Results are illustrated by examples.

This paper points out that input-to-state stability of zero dynamics having a continuously differentiable (instead of locally Lipschitz continuous) gain function suffices to guarantee the existence of globally stabilizing, smooth partial-state feedback control laws for cascade systems, without imposing any extra condition. This conclusion is proved via the small gain theorem and a novel variable separation technique combined with feedback domination design.

In most undergraduate texts on control systems, the Routh-Hurwitz criterion is usually introduced as a mechanical algorithm for determining the Hurwitz stability of a real polynomial. Unlike many other stability criteria such as the Nyquist criterion, root locus, etc. no attempt whatsoever is made to even allude to a proof of the Routh-Hurwitz criterion. Previous results using the Hermite Biehler theorem have, however, succeeded in providing a simple derivation of Routh's algorithm for determining the Hurwitz stability or otherwise of a given real polynomial. However, this derivation fails to capture the fact that Routh's algorithm can also be used to count the number of open right half plane roots of a given polynomial. This paper shows that by using appropriately generalized versions of the Hermite-Biehler theorem, it is possible to provide a simple derivation of the Routh-Hurwitz criterion which also captures its unstable root counting capability.

In this paper, we propose to define the transfer function of a system in a new way, namely as a semi-cancellable fraction (SCF), i.e. a fraction in which unstable pole-zero cancellations are not allowed. Within this new framework, it is possible to be rigorous and coherent in dealing with the stabilizability of a system from an input-output viewpoint, not focusing on the stabilizability of infinite dimensional input-output systems. >

The problem of robust design is regarded as a parameterization problem, and the choice of parameters to satisfy with respect to a class of bounded perturbations is considered. Application areas for the general results derived include multivariate root-locus robustness, design based on simplified or reduced plant models, and singular perturbation analysis. The results are based on asymptotic analysis of feedback structures and a proof of the existence of parameter ranges for robust stability. >

The determination of robust stability of interval polynomials with respect to an arbitrary closed convex contour in the left-half complex plane is considered. It is shown that the problem can be solved by checking only the root locations of a subset of the exposed edges of the family. The number of exposed edges to be checked depends on the damping margin defined by the contour and the degree of interval polynomials, but does not depend on the exact shape of the contour. A simple, easy-to-apply, graphical approach determines the subset of the exposed edges. An example illustrates the merit of the approach. >

Modified processes are proposed for directly treating singularities in the Routh array, and procedures are developed for determining the respective orders of simple and/or repeated roots lying on the imaginary axis. The extended Sturm test is developed for determining the root distribution on some specified lines. Moreover, to enhance the Routh stability theorem, the extended Routh theorem for finding the numbers of roots of a polynomial lying within some specified regions is proposed. >

Recently, it has been shown [8] that robust stability in model-reference adaptive control (MRAC) is guaranteed by simply using a static, instead of a dynamic, normalizing signal together with a parameter projection in the adaptive law. In this note, we show that the stability proof associated with such modification follows directly from the proof of schemes with dynamic normalization. Consequently, the proofs and stability arguments used in [8] can be simplified consider-ably. © 1993 IEEE 38 1 170 173 Cited By :2

A simple numerical procedure for estimating the stochastic robustness of a linear time-invariant system is described. Monte Carlo evaluation of the system's eigenvalues allows the probability of instability and the related stochastic root locus to be estimated. This analysis approach treats not only Gaussian parameter uncertainties but also nonGaussian cases, including uncertain but bound variations. Confidence intervals for the scalar probability of instability address computational issues inherent in Monte Carlo simulation. Trivial extensions of the procedure admit consideration of alternate discriminants; thus, the probabilities that stipulated degrees of instability will be exceeded or that closed-loop roots will leave desirable regions can also be estimated. Results are particularly amenable to graphical presentation. >

An approach is presented to the specification of optimal overshoot controllers when the controller order is fixed and the closed-loop system poles are at a fixed location. The essential effect of the technique is to pick the best possible locations of the free zeros of the system in terms of minimizing the overshoot. The solution is obtained by solving an affine minimax optimization problem. An example illustrates how this technique can be used as a design aid when it is necessary to trade off conflicting design requirements, such as overshoot and settling-time specifications. It is shown that the method is not restricted to the deadbeat case, but can also be used for any prescribed set of closed-loop poles. >

The problem of finding the conditions under which an interval polynomial has a given number of roots in the open left-half plane and the other roots in the open right-half plane, irrespective of the values of its coefficients, is considered. A simple criterion is provided to test interval polynomials for the root distribution invariance, viewed as an extension of Kharitonov's theorem. The goal is to provide an alternative theorem and then give an efficient means of checking the root distribution invariance. >

It is shown by example that the main theorem in the above paper by B.R. Barmish (see ibid., vol.34, p.157-65 (1989)) may lead to an erroneous D-stability conclusion for certain polynomials among the ones considered. Suggestions are presented and discussed. >

Classical control theory is concerned with the topics in our title in the context of single-input/single-output systems. There is now a large and growing literature on the extension of these ideas to the multiinput/multioutput case. This development has posed certain difficulties, some due to the intrinsic nature of the problem and some, we would argue, due to an inadequate reflection on what the multivariable problem calls for. In this paper we describe what seems to us to be the natural multivariable analogs of these concepts from classical control theory. A rather satisfactory generalization of the Nyquist criterion will be described, and a clear analog of the asymptotic properties of the root locus will be obtained in the "multiparameter" case. However, an example is given which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains. This calls for a new ingredient, a compactification of the space of gains, and perhaps the most interesting new feature in this circle of ideas comes in the area of pole placement. This problem is difficult in the multivariable case, but by establishing a correspondence with a classical set of problems in geometry, we are able to understand its main aspects and to derive results on pole placement by output feedback over the real field.

Frequency-domain methods are used to study the angles of arrival and departure for multivariable root loci. Explicit equations are obtained. For a special class of poles and zeros, some simpler equations that are generalizations of the single-input-single-output equations are presented.

This paper is concerned with single-loop, single-input-output, feedback control systems consisting of a plant, controller, and feedback sensor each modeled by a rational transfer function. The plant can be unstable, nonminimum phase, and even improper. The objective is the determination of the controller transfer function for which the system is asymptotically stable and design objectives are met. This has always been the goal in feedback system design. What is novel about the approach presented here is that the family of all stabilizing controller transfer functions is parameterized in terms of a rational function K (s) . The design method allows for complete freedom in the selection of all closed-loop system poles (except those associated with hidden modes). This is accomplished through the choice of the denominator polynomial for K (s) . The remaining design freedom lies in the selection of the numerator of K (s) . This freedom is utilized to shape the system sensitivity function and to achieve specified steady-state error coefficients. A numerical example in which the plant is unstable and nonminimum phase is presented to illustrate the above points. The final design is compared with alternative designs and the tradeoffs are clearly displayed.

Considers the monic polynomial f(z):=z/sup n/+a/sub n-1/z/sup n-1/+. . .+a/sub 0/ in the complex variable z with complex coefficients. Under the assumption that the nonleading coefficients of f lie in the disk mod z mod >

The paper adapts the classical root locus plot analysis to some one-dimensional, controlled, distributed, viscoelastic structures. The constitutive law that is used for the viscoelastic material may be modeled with fractional order derivatives provided the control law(s) involve similar derivatives. Two sample one-dimensional structures and control systems are used to introduce the approach. A partial differential equation representation of each structure is used as well as a Ritz approximation. The effect of error due to approximation upon stability conclusions is discussed in the context of this comparison.

An efficient computer-aided root-locus method is described The approach is based on the concept of continuation methods in which the solution of a parameterized family of algebraic problems is converted into the solution of a differential equation. The root-locus plot is obtained in a systematic manner by numerical integration. Singularities are analyzed and classified according to the properties of higher order derivatives. Depending on their classification, singular points on the root loci are taken care of accordingly.

While the Pade approximation can often be used to produce good reduced-order transfer functions, the method suffers from the disadvantage that a stable reduced order-model is not always obtained for an original model that is stable. A method is described for obtaining stable partial Pade approximation reduced-order models. The method permits the determination of the effect of changes in the coefficients in the partial Pade approximation on the poles of the reduced-order transfer function by means of standard root locus plots.

Sufficient conditions are presented for the existence of periodic motions in a class of autonomous, time-invariant, non-linear feedback systems. The conditions can be interpreted as circle type criteria, stated in terms of the frequency response and root locus diagrams for the system's linear part, and in terms of the characteristics of the nonlinearity.

A formula is given for the closed-loop root locus of a linear control system, designed using the separation principle, when the loop gain is changed from its nominal value. An illustrative example is given, and extensions to dynamic mismatch, discrete-time systems, and reduced-order dynamics are discussed.

A new method of plotting the root-loci is developed for the linear control system with delay in control or in state. In case of the system with delay in control, the root-locus plot starts from neighborhoods of the open-loop zeros instead of the open-loop poles and thus the effect of time-delay is easily handled. In case of the system with delay in state, the open-loop poles are firstly found by applying the root-locus method for the system with delay in control and then the desired root-loci are found by starting the root-loci plot from the open-loop poles.

A method is developed for determining the stability region in the loop-gain space for a linear multivariable control system. Two new interpretations of the return difference determinant are employed, allowing the stability region boundaries to be determined accurately by means of an algorithm.

Linear time-invariant systems with uncertain parameters are considered. A test is given which is both necessary and sufficient for root clustering of a family of polytopic polynomials in a simply connected domain. The test does not require checking a continuum of polynomials. Special cases include stability testing of continuous-time systems and of discrete-time systems with uncertain parameters, as well as 'relative stability' tests of such systems. Interval polynomials can also be handled as a special case. >

We study the asymptotic behavior of the dosed loop eigenvalues (root loci) of a strictly proper linear time-invariant control system as loop gain goes to \infty . The formulas are stated in terms of the eigenvalues of nested restricted linear maps of the form A(\mod S_{2})|_{S_{1}} where S 1 and S 2 are subspaces of complementary dimension. Additional geometrical insight into the formulas is obtained by mechanizing the formulas using orthogonal projections. Our method and formulas are useful in other asymptotic calculations as well, e.g., hierarchical multiple-time scales aggregation of Markov chains with some infrequent transitions.

A quasi-dipole is defined to be a pole-zero pair embedded in a suitable approximation to a linear phase field. When complex conjugate plant poles occur near the stability boundary, quasi-dipoles often result from controller designs having compensating zeros. Closed-form root loci expressions are developed for quasi-dipoles; these expressions are shown to provide excellent root locus approximations for practical problems.

A general compensation design algorithm for pole assignment is considered. The methodology presented is an extension of previous work [10] in which plant and compensator dynamics are described in such a way that the influence of compensation on the poles of the system is directly observed. By holding some of the poles of the plant and compensator stationary at each iteration, the design problem is transformed into an equivalent multi-input, single-output system to which root locus techniques can be directly applied. The design algorithm is iterative in nature and allows the designer the freedom to use his expertise within a standardized framework. An example of a general compensation problem is given to illustrate the simplicity of the design method.

The graphical method of constructing root locus is very well known. The analytical methods of obtaining root locus equations developed in the Soviet Union merit utilization due to the flexibility inherent in the manipulation of algebraic expressions. Arrays are developed whereby these algebraic expressions can be obtained from a knowledge of the open loop transfer function. Breakaway points on the real axis and intersection with the imaginary axis can be obtained directly from these arrays. Finally, arrays are formulated for the case where one of the parameters of the open loop transfer function varies.

The objective of this paper is to present a frequency-domain stability criterion for pulse-width-modulated feedback systems. The criterion states that 1) if the Nyquist locus of the pulse transfer function of the linear plant does not intersect or encircle a certain disk in the complex plane, then the system is l 2 bounded input-bounded output stable, and 2) if the impulse response of the linear plant is zero at t = 0 , then the radius of the disk approaches zero as the ratio of the linear-plant bandwidth to the pulse-width-modulator frequency approaches zero. It is shown that the result in 2) implies that the criterion in 1) essentially becomes the well-known Nyquist criterion for sample-data systems when the pulse-width-modulator frequency is sufficiently large. This result permits the use of the standard design techniques for linear sample-data systems in a wide class of pulse-width-modulated systems. Consequently the design of these systems can be greatly simplified.

The goals of this short paper are twofold. First, the parameter stability region, or the D -decomposition technique as it is called in the Russian literature, for obtaining regions of stability in terms of one complex or two real parameters is shown to be applicable when certain elements of the system are given by numerical frequency response data. Second, a technique is presented for interpreting the parameter regions of stability obtained by the D -decomposition method in a space of several parameters.

The paper investigates the stability of feedback systems with one nonlinear element whose input-output characteristic is contained within a sector of the input-output plane. The results are applicable to any nonlinearity within a given sector and the conditions for global asymptotic stability are necessary if the exact analytic description of the nonlinearity is not available. The conditions for asymptotic stability are related to the Routh-Hurwitz and Nyquist stability criteria for linear systems.

A method of describing a system in parameter space is presented in this paper. This method can be applied to the analysis and design of high-order control systems with multiple variable or adjustable parameters.

In this paper a rigid missile (booster) guidance and control system is synthesized by means of an advanced control optimization technique called the Parametric Expansion Method. It was developed by Dr. Charles W. Merriam III of the General Electric Research Laboratory, Schenectady, N. Y. The results obtained are evaluated in a digital simulation of the missile dynamics and compared with those obtained using a similar system which was synthesized by the more conventional means of root locus.

A method is presented whereby the absolute and relative stability of linear control systems containing transport lag or distributed lag can be determined. The method utilizes a variation of the generalized parameter plane technique as developed by D. D. Siljak and all results are in terms of two free system parameters. An example of a particular system with transport lag is analyzed. In the example, the system stability is analyzed with respect to the simultaneous variation of a controller gain and time constant.

A stability criterion for feedback systems with a linear time-invariant forward loop and a periodically time-varying gain in the feedback loop is developed. As most recent frequency-domain stability criteria, it requires the existence of a multiplier having certain properties. A graphic technique is given to determine whether or not this multiplier exists. The proof of the criterion is based on the Floquet theory and the classical theory of Toeplitz forms. Examples to illustrate the criterion are included.

A frequency domain stability criterion is given for a feedback system whose loop consists of an instantaneous nonlinear amplifier, a linear dynamical system, and a transducer with backlash. A class of input signals is considered that essentially consists of constant-amplitude functions and exponentially decaying functions. Separate considerations are given to the case when the system includes a saturating instantaneous amplifier and the case when the nonlinear amplifier is not a saturating type. In both cases a single criterion guarantees the stability. A few examples are given to illustrate the proposed stability criterion. Finally it is shown that the stability criterion can be applied to a large class of autonomous non-linear feedback systems containing a nonlinearity with dead-zone or with hysteresis and dead-zone.

A frequency domain criterion is formulated to prove global asymptotic stability of feedback control system with more than one nonlinear element. The input-output characteristics of the nonlinearities are confined to sectors of the input-output planes. Sufficient conditions for global asymptotic stability are derived in terms of a number of inequalities equal to the number of nonlinear elements.

The purpose of this paper is to establish the stability of single-loop feedback systems with one differentiable nonlinear element. In those cases where the Popov criterion fails to guarantee stability for the entire sector predicted by Aizerman's conjecture, new results can be obtained by restricting the slope of the nonlinear function. A new frequency domain stability criterion is obtained which, like the Popov criterion, has only one unknown parameter. Thus, a simple graphical interpretation is possible. Examples are given which show a considerable improvement over the Popov criterion.

An extension of Horowitz's root-locus compensation design method is presented for analyzing and designing a high-gain linear system having variable coefficients. Rules for positioning the far-out poles of such a system based upon known specifications are established. These rules, when applicable, permit a high-gain system capable of handling large variations in the loop gain parameter to be successfully designed.

This paper shows a method for using describing functions in a control system with multiple nonlinearities. The linear system resulting after the proper describing functions have been inserted is analyzed using an "input dependent root locus" which is a modified root locus technique. The rules for construction of the input dependent locus and a proof of its applicability are given along with examples which illustrate the method. The technique shows the existence of limit cycles, their amplitude and frequency when they exist, and the stability of the limit cycle.

A direct synthesis method is presented for obtaining a discrete compensator to produce a finite-settling-time transient in response to a prescribed input with no intersample transient terms after the finite settling time. The method is applicable to a single constant-frequency sampler in a feedback system with any type of hold for any polynomial input. The method uses a fictitious sampler at the output with sample time T/m to scan the intersample times and to obtain criteria for deadbeat transients. The criteria are derived from root locus considerations but are applied to Laplace transforms containing both functions of s and functions of z so that the only operations are analytic ones.