Search

This work presents generalized forgetting recursive least squares (GF-RLS), a generalization of RLS that encompasses many extensions of RLS as special cases. First, sufficient conditions are presented for the 1) Lyapunov stability, 2) uniform Lyapunov stability, 3) global asymptotic stability, and 4) global uniform exponential stability of parameter estimation error in GF-RLS when estimating fixed parameters without noise. Second, robustness guarantees are derived for the estimation of time-varying parameters in the presence of measurement noise and regressor noise. These robustness guarantees are presented in terms of global uniform ultimate boundedness of the parameter estimation error. A specialization of this result gives a bound to the asymptotic bias of least squares estimators in the errors-in-variables problem. Lastly, a survey is presented to show how GF-RLS can be used to analyze various extensions of RLS from literature.

Lur’e systems are dynamical systems that are characterized by the feedback interconnection between a linear, time-invariant system and a feedback nonlinearity. Lur’e systems have been used to characterize the dynamics of several systems including gas turbine combustors and self-oscillatory systems. In this paper, we introduce an Identification algorithm for Lur’e systems with hysteretic feedback nonlinearities. We assume that the input to the Lur’e system is an unknown constant signal, and the linear dynamics have nonzero initial conditions. First, we use least squares with a transfer function model to identify the linear dynamics of the Lur’e system. Then, we use the identified linear dynamics along with the measured output to construct an estimate of the hysteretic nonlinearity. We show numerical examples to illustrate the proposed approach.

Thermoacoustic systems are self-oscillating due to the fact that a constant input (e.g., fuel rate in gas-turbine combustors) yields an asymptotically oscillatory response. This behavior arises from the interaction between combustion and acoustics, resulting in thermoacoustic oscillations. Under varying operating conditions, the dynamics of thermoacoustic systems may change dramatically, which may require that active suppression controllers can be tuned for specific working conditions. This work provides an experimental investigation of retrospective cost adaptive control for suppressing thermoacoustic oscillations under sampled-data control and varying system operating conditions. This approach is first applied to a Rijke-tube emulation model for hyperparameter selection and subsequently to an experimental Rijke-tube setup. Physical experiments are conducted to investigate the performance and robustness of the adaptive controller under varying operating conditions.

One of the most striking phenomena in systems and control theory is initial undershoot, where the initial direction of the response of a system to a step input or setpoint command is opposite to the direction of the asymptotic response. As mentioned in “Summary,” initial undershoot represents a fundamental limitation on the performance of a control system, and it has potentially serious consequences, especially when the initially “wrong” direction violates constraints on the output of the system. For example, in motion control applications, such as robotics or autonomous vehicles, initial undershoot may result in a collision. In any event, initial undershoot is an intriguing, visual example of a system-theoretic phenomenon that occurs in electrical, mechanical, and cyberphysical systems, and it is one of the many deleterious effects of nonminimum-phase zeros on the achievable performance of feedback control systems [1]–[3].

Among the multitude of modern control methods, model predictive control (MPC) is one of the most successful [1]–[4]. As noted in “Summary,” this success is largely due to the ability of MPC to respect constraints on controls and enforce constraints on outputs, both of which are difficult to handle with linear control methods, such as linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG), and nonlinear control methods, such as feedback linearization and sliding mode control. Although MPC is computationally intensive, it is more broadly applicable than Hamilton–Jacobi–Bellman-based control and more suitable for feedback control than the minimum principle. In many cases, the constrained optimization problem for receding-horizon optimization is convex, which facilitates computational efficiency [5].

The objective of navigation is to determine the instantaneous location of a moving vehicle. Historically, sailors estimated the speed of a ship by dropping a log into the water and observing the time for it to float from the bow to the stern [1, pp. 21-38]. Combining this speed information with directional data from a compass yields a velocity estimate. Finally, integrating the velocity estimate with respect to time and using the known initial position yields a position estimate at each instant of time. This procedure is called dead reckoning [2, p. 30], [3] or, when performed by animals without the benefit of calculus and digital computation, path integration [4]. Although not all navigation is based on dead reckoning-for example, celestial navigation uses the stars for position fixing-it is widely used due to its ability to utilize speed, direction, and time data from a variety of sensors. By computing position from velocity data, dead reckoning is an integration process. It is important to stress, however, that, because of the lack of observability, dead reckoning cannot take advantage of an observer and, thus, cannot stabilize the integrator dynamics. Therefore, dead reckoning is an inherently unstable process, regardless of how the velocity and direction data are acquired.

Although realizations of transfer functions are a standard topic in textbooks about linear systems theory, this article focuses on several points that are not widely treated. First, it is shown that, for a given dynamics matrix A, the smallest number of sensors and actuators that can be used to form a minimal realization is determined by properties of the Jordan form of A. This leads to the notions of minimally sensed and minimally actuated systems. Next, the rank of a transfer function is related to properties of the Jordan form of A and the rank of the Rosenbrock system matrix.

The ability to estimate parameters depends on two things: identifiability [1] (which is the ability to distinguish distinct parameters) and persistent excitation (which refers to the spectral content of the signals needed to ensure convergence of the parameter estimates to the true parameter values) [2]-[4]. Roughly speaking, the level of persistency must be commensurate with the number of unknown parameters. For example, a harmonic input has 2D persistency and thus can be used to identify two parameters, whereas white noise is sufficiently persistent for identifying an arbitrary number of parameters. Within the context of adaptive control, persistent excitation is needed to avoid bursting [5]; recent research has focused on relaxing these requirements [6]-[8].

Although controllability and observability are distinct properties, one of the fundamental-and most attractive-results of our field is the fact that (A, B) is controllable if and only if f (AT,BT) is observable. This duality provides a deep linkage between the linear-quadratic regulator (LQR), which seeks a feedback gain K such that A + BK is asymptotically stable, and the linear-quadratic estimator (LQE), which seeks an output-error-injection gain F such that A+FC is asymptotically stable. In the case of LQR, the controllability of (A, B) implies that there exists a feedback gain K that arbitrarily places the eigenvalues of A+BK, thus facilitating closed-loop asymptotic stability. In the dual case of LQE, the observability of (A, C) implies that there exists an error-injection gain F that arbitrarily places the eigenvalues of A + FC, thus facilitating closedloop asymptotic stability of the error dynamics. A key distinction worth noting is that A + BK is the dynamics matrix of a physical feedback loop, whereas A + FC is the dynamics matrix of a nonphysical error system.

Although the Kalman filter is often presented within a continuous-time context [1]-[4], the original derivation was provided in discrete time [5]. In practice, controllers and observers are invariably implemented digitally; thus discrete-time algorithms deserve special attention. This article focuses on the real-time implementation of the discrete-time Kalman filter and its relation to the discrete-time Kalman predictor. These algorithms differ in subtle ways, as highlighted in this article. Equations for the discrete-time Kalman filter and predictor are given in [6]. Unfortunately, the covariance matrix becomes indefinite after a few steps, which shows that those equations are erroneous. The present article corrects and extends the results from [6].

Power transmission matrices (PTMs) for structures are 2 X 2 matrices that relate force and velocity at one terminal to force and velocity at another terminal. This technique provides an elegant approach to deriving transfer functions for structures consisting of masses, springs, and dashpots, that is, dampers. PTMs were developed during the 1970s and 1980s, but they are rarely mentioned today except briefly in textbooks. The first goal of this tutorial is to highlight the utility of this modeling technique. A related aim is to bring inerters into this framework. The article revisits the classical topic of analogies by deriving PTMs for twoport networks. The across-through analogy is used to construct circuits whose transfer functions match those of the analogous structure. This article is intended for all students and practitioners of control systems who may benefit from awareness of this modeling technique.

Control in the power distribution system is needed to match the generation of electricity (that is, supply) with the demand for power (that is, load) at all times while maintaining a high level of power quality [1] (see "Summary"). The complexity of controlling the electric distribution system increases when distributed intermittent energy sources, such as solar photovoltaics (PVs), displace large-scale dispatchable generation [2]-[10]. If the supply from distributed resources exceeds the time-varying demand across a feeder, then the feeder-level voltage will increase. In extreme cases, large voltage increases will trigger the disconnection of distributed generation to protect electrical equipment and ensure customer safety. Consequently, coordinating renewable energy sources that are distributed throughout the distribution system-to match real-time demand across an entire power network-brings the additional challenge of maintaining a high level of supply reliability.

An algebraic approach to the synthesis of a dynamic system that reconstructs the generic inaccessible input of a discrete-time linear multivariable system with unknown initial state is discussed. The method devised exploits geometric properties of key subspaces for the original system and algebraic properties of the Moore-Penrose inverse of Toeplitz matrices related to the algorithms for computing those subspaces. Nonminimum-phase invariant zeros are taken into account implicitly with the proposed techniques, while minimum-phase invariant zeros require that a filter be inserted between the original system and the reconstructor. The procedure applies to either strictly-proper or non-strictly-proper systems.

This paper presents a two-step method for identification of the SISO Hammerstein model, which employs input autocorrelation and input-output cross-correlation functions as data for least-squares estimation. Using separable processes as input signals, the proposed method allows the linear block of a Hammerstein model to be identified up to a multiplicative constant, without a prioriknowledge of the nonlinear model structure. Both ARX and μ-Markov structures of the linear block are considered, where the main concern is the accuracy of pole and zero estimates. The correlation least-squares method is compared numerically with a well-known nonlinear least-squares method, which shows that the correlation method is consistently accurate across different nonlinear model structures.

We consider reduced‐order and subspace state estimators for linear discrete‐time systems with possibly time‐varying dynamics. The reduced‐order and subspace estimators are obtained using a finite‐horizon minimization approach, and thus do not require the solution of algebraic Lyapunov or Riccati equations. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Really good question. Actually, someone asked me that before, but I never got around to looking into it.

In this paper we consider a semilinenr Duhem model. The input-output map of the model is rate-independent, thus yielding persistent phase shift (that is. hysteresis) at arbitrarily low frequency. For the semilinear Duhem model we reparameterize the response in terms of the control input, and we provide sufficient conditions for convergence to a hysteresis map. A constrained least squares method is developed to identify the hysteresis map using the semilinear Duhem model.

The problem of a rigid body tracking a desired angular velocity trajectory is addressed using adaptive feedback control. An adaptive controller is developed for a planar rotating body tracking a desired angular velocity command. Lyapunov analysis is used to show that tracking is achieved globally. A periodic angular velocity command is then used to identify the inertia parameter. The adaptive controller is implemented on a triaxial attitude control testbed with fan thrusters. A piecewise linear approximation of an observed input non linearity is inverted to obtain improved angular velocity tracking and inertia identification. To eliminate residual tracking error, an adaptive algorithm is used for improved feedback linearization. Lyapunov analysis is used to show boundedness of the angular velocity and inertia estimate errors. The approach is validated by numerical simulation.

In this paper we develop a discrete-time adaptive stabilization algorithm based on a one-step backward-horizon cost criterion. By optimizing the cost with respect to the update step size, we obtain a gain update law that guarantees convergence of the plant states. The convergence proof is based on a modified Lyapunov technique. We extend the algorithm to include integral control for rejecting constant disturbances and we present an experimental application to DC motor positioning system. Copyright © 2003 John Wiley & Sons, Ltd.

In this paper we develop a discrete‐time adaptive stabilization algorithm based on a one‐step backward‐horizon cost criterion. By optimizing the cost with respect to the update step size, we obtain a gain update law that guarantees convergence of the plant states. The convergence proof is based on a modified Lyapunov technique. We extend the algorithm to include integral control for rejecting constant disturbances and we present an experimental application to DC motor positioning system. Copyright © 2003 John Wiley & Sons, Ltd.

In this paper, we formulate robust stability and performance bounds in terms of guaranteed cost inequalities. We derive new guaranteed cost bounds for plants with real structured uncertainty, and we reformulate them as linear matrix inequalities (LMIs). In particular, we obtain a shifted linear bound and a shifted inverse bound, and give LMI forms for a shifted bounded real bound, a shifted Popov bound, a shifted linear bound and a shifted inverse bound. Several examples are used to compare the shifted bounds with their unshifted counterparts and to make comparisons among these new bounds and vertex LMI bounds. Copyright © 2002 John Wiley & Sons, Ltd.

Abstract.: In this paper we develop explicit formulas for induced convolution operator norms and their bounds. These results generalize established induced operator norms for linear dynamical systems with various classes of input–output signal pairs.

This paper considers fixed‐structure stable ℋ︁2‐optimal controller synthesis using a multiobjective optimization technique which provides a trade‐off between closed‐loop performance and the degree of controller stability. The problem is presented in a decentralized static output feedback framework developed for fixed‐structure dynamic controller synthesis. A quasi‐Newton/continuation algorithm is used to compute solutions to the necessary conditions. To demonstrate the approach, two numerical examples are considered. The first example is a second‐order spring–mass–damper system and the second example is a fourth‐order two‐mass system, both of which are considered in the stable stabilization literature. The results are then compared with other methods of stable compensator synthesis. Copyright © 2000 John Wiley & Sons, Ltd.

This paper considers fixed-structure stable ℋ2-optimal controller synthesis using a multiobjective optimization technique which provides a trade-off between closed-loop performance and the degree of controller stability. The problem is presented in a decentralized static output feedback framework developed for fixed-structure dynamic controller synthesis. A quasi-Newton/continuation algorithm is used to compute solutions to the necessary conditions. To demonstrate the approach, two numerical examples are considered. The first example is a second-order spring–mass–damper system and the second example is a fourth-order two-mass system, both of which are considered in the stable stabilization literature. The results are then compared with other methods of stable compensator synthesis. Copyright © 2000 John Wiley & Sons, Ltd.

This is a humorous look at papers for linear time-invariant systems.

In this paper, we present a Lyapunov function for systems containing a double integrator and with controller saturation. This Lyapunov function is composed of a positive‐semidefinite quadratic term and an integral term. The main result provides a sufficient condition that guarantees a system with a double integrator can be globally stabilized by a saturating linear controller. For a triple‐integrator system the saturated linear controller does not satisfy the sufficient condition, which agrees with the known result. Copyright © 1999 John Wiley & Sons, Ltd.

In this paper, we present a Lyapunov function for systems containing a double integrator and with controller saturation. This Lyapunov function is composed of a positive-semidefinite quadratic term and an integral term. The main result provides a sufficient condition that guarantees a system with a double integrator can be globally stabilized by a saturating linear controller. For a triple-integrator system the saturated linear controller does not satisfy the sufficient condition, which agrees with the known result. Copyright © 1999 John Wiley & Sons, Ltd.

Real-time simulation often requires a computer to interact with external hardware in a dynamic loop. Data must be sampled from input signals and incor porated into the numerical integration algorithm to evaluate the derivatives of the state variables. Also, data from the integration must be available for external use with a minimum of delay. This paper will show that, because of these requirements, cer tain numerical integration routines are best suited for real-time simulation. A new fourth-order Runge- Kutta routine designed for real-time use is presented along with its derivation.

This paper describes and illustrates a practical methodology for robust, fixed-structure controller synthesis using a decentralized static output feedback problem formulation. To account for real-parameter structured uncertainty, scaled Popov bounds for the real structured singular value are employed. A worst-case H2cost bound with respect to the controller parameters is formed, and quasi-Newton optimization algorithms are used to solve the numerical optimization problem. Numerical results are provided for a MIMO 16th-order acoustic duct model with uncertain dynamics for both centralized and decentralized controller structures.

Hamilton-Jacobi-Bellman theory is shown to provide a unified framework for nonlinear feedback control laws for special classes of nonlinear systems. These classes include Jurdjevic-Quinn type systems, as well as minimum phase systems with relative degree {1, 1, ..., 1}. Several examples are given to illustrate these results. For the controlled Lorenz equation, results obtained by Vincent and Yu are extended. Next, for spacecraft angular velocity stabilization with two torque inputs, a family of nonlinear feedback control laws that globally asymptotically stabilize angular velocity is established. Special cases of this family of control laws include generalizations of the locally stabilizing control laws of Brockett and Aeyels to global stabilization as well as the globally stabilizing control laws of Irving and Crouch and Byrnes and Isidori. Finally, the results are applied to spacecraft angular velocity stabilization with only one torque input. These last results extend control laws given by Outbib and Sallet.

A model-reduction problem is considered which involves both L2 (quadratic) and H∞ (worst-case frequency-domain) aspects. Specifically, the goal of the problem is to minimize an L2 model-reduction criterion subject to a prespecified H∞ constraint on the model-reduction error. The principal result is a sufficient condition for characterizing reduced-order models with bounded L2 and H∞ approximation error. The sufficient condition involves a system of modified Riccati equations coupled by an oblique projection, i.e. idempotent matrix. When the H∞ constraint is absent, the sufficient condition specializes to the L2 model-reduction result given by Hyland and Bernstein (1985).

A technique for synthesizing controllers that minimize an upper bound for the worst-case H2norm over a set of real, parametric uncertainty is presented. The parameter uncertainty may appear simultaneously in the state, input, and output matrices of the plant's state space realization. The scaled Popov robust stability criterion is used to derive an upper bound for the worst-case H2norm. Necessary conditions for the controller gains that minimize this upper bound are derived. Numerical examples demonstrate the improvement in robust performance of the robust controllers over LQG controllers.

This paper presents three novel shifted quadratic guaranteed cost bounds for analysis and synthesis of robust feedback controllers. Numerical examples are given to compare the robust stability and performance tradeoffs of these controllers with controllers based upon scaled Popov synthesis.

A symbolic matrix differentiation routine called MatGrad is developed for computing gradient expressions for fixed-structure controller synthesis. These expressions can be used with gradient-based optimization algorithms to compute controllers that meet H2, H∞, μ and Pole placement objectives in both continuous and discrete time. MatGrad is written in Mathematica and utilizes routines from the noncommutative algebra package NCAlgebra developed by Helton and Miller.

The optimal projection equations for discrete-time reduced-order dynamic compensation are generalized to include the effects of state-, control- and measurement-dependent noise. In addition, the discrete-time static output feedback problem with multiplicative disturbances is considered. For both problems, the design equations are presented in a concise, unified manner to facilitate their accessibility for developing numerical algorithms for practical applications.

This paper fully describes the RTAC experimental testbed which provides a means for implementing and evaluating nonlinear controllers. The description of the testbed includes all physical parameters of the device that are relevant to the design and implementation of controllers. Next, four nonlinear controllers are considered. The first controller is a static, full‐state‐feedback, globally asymptotically stabilizing control law developed using partial feedback linearization and integrator backstepping. Next, three nonlinear controllers based upon passivity principles are presented, two of which are encompassed by the classical passivity framework, while the third is based upon the novel concept of virtual resetting absorbers. The passive controllers do not require either translational position or velocity measurements. All of the controllers are implemented on the RTAC testbed and their performance is examined. © 1998 John Wiley & Sons, Ltd.

This paper describes and illustrates a unified methodology for robust, fixed‐structure controller synthesis. The approach is based upon direct fixed‐structure controller synthesis using a decentralized static output feedback formulation as a general framework for representing a large class of controller structures. Scaled Popov bounds for the real structured singular value are used to account for real parameter uncertainty and provide the means for optimizing a worst‐case ℋ︁2cost bound with respect to the free parameters of the controller. Quasi‐Newton optimization algorithms are used to solve the resulting numerical optimization problem. Initial stability multiplier and scaling matrices needed in scaled Popov synthesis are obtained by solving an LMI feasibility problem. Using both centralized and decentralized controller structures, numerical results are obtained for a 16th‐order acoustic duct model with uncertain damped natural frequencies and for a two‐dimensional beam‐spring model with uncertain actuator locations. © 1998 John Wiley & Sons, Ltd.

The oscillating eccentric rotor has been widely studied to model resonance capture phenomena occurring in dual-spin spacecraft and rotating machinery. This phenomenon arises during spin-up as a resonance condition is encountered. We consider the related problem of rotor despin. Specifically, we determine nonlinear feedback control laws that not only despin the rotor but also bring its translational motion to rest. These globally asymptotically stabilizing control laws are derived using partial feedback linearization and integrator backstepping schemes. For the case in which the oscillating eccentric rotor is excited by a translational sinusoidal forcing function, the control law is shown to attenuate the amplitude of the translational oscillation.

We consider the problem of controlling a top to the sleeping motion using two different actuation schemes. For a fixed‐base top two actuators are assumed to provide forces at the centre of mass in inertially fixed directions, while for a cart‐mounted top two actuators are assumed to apply forces to the cart in inertially fixed directions. The controller for the cart‐mounted top is obtained from the controller designed for the fixed‐base top using d'Alembert's principle. Both controllers are proved to be globally asymptotically stabilizing. For the uncontrolled fixed‐base top, necessary and sufficient conditions for Lyapunov stability of the sleeping motion are derived. For the case in which there is only one force actuator, locally asymptotically stabilizing control laws that drive the fixed‐base top to the sleeping motion are also obtained.

The acoustics and vibration of coupled structures can be efficiently modeled by means of energy flow theory, usually known as statistical energy analysis (SEA). SEA is based upon relations derived from the interaction of pairs of coupled modes and is applied empirically to coupled structures possessing multiple interacting modes. This talk investigates the accuracy of SEA models by deriving exact energy flow relations based upon modal models. Using white noise disturbances but otherwise deterministic structural models, we present energy flow relations for multiple coupled structures. These relations are formulated in terms of thermodynamic energy, which is a measure of subsystem energy dissipation relative to subsystem disturbance intensity. The resulting energy flow models are compared to SEA models based upon coupled, uncoupled, and blocked subsystem energy. The results are applied to systems of coupled oscillators as well as coupled beams to demonstrate predicted energy flow under both strong and weak coupling.