Showing papers by "Dennis S. Bernstein published in 2016"

A tutorial and overview of retrospective cost adaptive control

Abstract: The goal of this paper is to provide a tutorial on retrospective cost adaptive control (RCAC). RCAC is a discrete-time adaptive control technique that is applicable to stabilization, command following, and disturbance rejection. RCAC is based on the concept of retrospectively optimized control, where past controller coefficients used to generate past control inputs are re-optimized in the sense that if the reoptimized coefficients had been used over a previous window of operation, then the performance would have been better. Unlike signal processing applications such as estimation and identification, it is impossible to change past control inputs, and thus the re-optimized controller coefficients are used only to generate the next control input. This paper presents a tutorial on the algorithmic details of RCAC including the construction of the retrospective cost, the role of the target model, and the effect of tuning parameters. Numerical examples are given to illustrate each of these choices as well as the performance of RCAC for command following and disturbance rejection under minimal modeling of the plant dynamics and exogenous inputs. Properties of the closed-loop system are also compared to features of discrete-time, high-authority LQG controllers.

Aircraft sensor fault detection using state and input estimation

Abstract: This paper presents a method for detecting aircraft sensor faults using state and input estimation. We formulate the kinematics as a nonlinear state space system, which requires no modeling information, and thus is applicable to all aircraft. To illustrate the method, we investigate three fault-detection scenarios, namely, faulty pitot tube, angle-of-attack sensor, and accelerometers. We use the extended Kalman filter for pitot-tube and angle-of-attack sensor fault detection, and retrospective cost input estimation for accelerometer fault detection. For numerical illustration, we use the NASA Generic Transport Model to detect stuck, bias, drift, and deadzone sensor faults.

Maneuvering target tracking using retrospective-cost input estimation

Abstract: This paper uses a Kalman filter with retrospective cost–based input estimation (KF/RCIE) to track maneuvering targets with unknown acceleration. Unlike conventional tracking methods that model the acceleration as a random process, KF/RCIE views the unknown acceleration as a deterministic unknown signal. Retrospective cost optimization is then used to estimate the unknown acceleration. Numerical examples and laboratory experiments illustrate the effectiveness of this approach with comparison to conventional tracking methods.

Experimental application of transmissibility operators to fault detection

Abstract: This paper considers a technique for fault detection called sensor-to-sensor identification. Sensor-to-sensor identification takes advantage of freely available and unknown external (ambient) excitation to identify a sensor-to-sensor model (i.e., a transmissibility operator), which is independent of the excitation signal and the initial conditions of the underlying system. In the presence of unknown external excitation, the identified transmissibility operator is used to compute the sensor-to-sensor residual, which is the discrepancy between the predicted sensor output (based on the transmissibility operator) and the actual measurements. The sensor-to-sensor residuals are used to detect and diagnose faults in sensors and system dynamics. We consider an experimental setup consisting of a drum with two speakers and four microphones. Each speaker is an actuator, and each microphone is a sensor that measures the acoustic response at its location. Measurements from the four microphones are used to construct transmissibility operators, which in turn are used to detect changes in the dynamics of the drum by computing the resulting one-step residual.

On the stability and convergence of a sliding-window variable-regularization recursive-least-squares algorithm

Abstract: A sliding-window variable-regularization recursive-least-squares algorithm is derived, and its convergence properties, computational complexity, and numerical stability are analyzed. The algorithm operates on a finite data window and allows for time-varying regularization in the weighting and the difference between estimates. Numerical examples are provided to compare the performance of this technique with the least mean squares and affine projection algorithms. Copyright © 2015 John Wiley & Sons, Ltd.

An experimental evaluation of the forward propagating Riccati equation to nonlinear control of the Quanser 3 DOF Hover testbed

Abstract: This study presents an experimental evaluation of the forward-propagating Riccati equation (FPRE) control. FPRE employs a state-dependent coefficient (SDC) parameterization of the nonlinear dynamics, and the feedback gains are updated in real time. The efficacy of the proposed control algorithm is verified by experimental studies on the Quanser 3 DOF Hover system. The ability of FPRE to follow the desired references is investigated, and its performance is compared with the conventional linear-quadratic regulator. Experimental results show the effectiveness of FPRE for following given references in the considered operating envelope.

Parameter estimation in the Burgers equation using retrospective-cost model refinement

Abstract: We apply retrospective cost model refinement to parameter estimation in a nonlinear partial differential equation. Specifically, for the scalar Burgers equation, we estimate the viscosity from measurements of flow velocity at a single grid point. We also consider the analogous problem for a modified Burgers equation as a proxy for a large eddy simulation to estimate a parameter that relates subgrid-scale stresses to the resolved strain rate.

Laurent Series and ℓp Sequences

Abstract: Proof. To show i) implies ii), since the Laurent series is absolutely convergent in P [1, p.168], setting z = 1 implies ∑∞ i=0 |gi | < ∞. Hence, (gi)∞ i=0 ∈ 1 ⊂ p for all p ∈ [1, ∞). To show iii) implies i), let p ≥ 1. By [2, p. 277], g has a minimal realization A ∈ Rn×n, B ∈ Rn, C ∈ R1×n, D ∈ R, where n is the number of poles of g, such that g(z) = C(z In − A)−1 B + D, where In is the n × n identity matrix. The Laurent series of g in P is g(z) = ∑∞ i=1 C Ai−1 Bz−i + D, and thus, for all i ≥ 1, gi = C Ai−1 B. Since (A, B, C) is minimal, the n × n observability and controllability matrices

Adaptive input estimation for nonminimum-phase discrete-time systems

Abstract: The accuracy of state estimation can be enhanced by simultaneously estimating unknown inputs. This paper presents an extension of retrospective cost input estimation (RCIE) that directly updates the estimates of all states. We show that RCIE can be used for systems in which the transmission zeros from the estimated input to the measurement are nonminimum phase. We demonstrate this ability on numerical examples, and we compare the estimates from RCIE to estimates from prior methods for input estimation. Finally, we use this technique to estimate the acceleration of a flight vehicle using camera data, and we assess the accuracy of the acceleration estimates by transforming the onboard body-frame acceleration measurements to the camera frame.

Experimental investigation of spatial spillover in adaptive feedback noise control of broadband disturbances in a 3D acoustic space

Abstract: Active feedback noise control for rejecting broadband disturbances must contend with the Bode integral constraint, which implies that suppression over some frequency range gives rise to amplification over another range. This is called spectral spillover. In the present paper, we apply retrospective cost adaptive control (RCAC) to active noise suppression in the interior of an automobile. This study highlights spectral spillover along with related issues, such as the need for controller stability, the effect of plant rolloff at DC, and the challenge of nonminimum-phase zeros. Beyond these issues, this paper deals with spatial spillover, which refers to the amplification of noise at locations where no microphone is located. Typically, this issue is dealt with by restricting the bandwidth to a frequency range within which the acoustic wavelength is sufficiently large such that the phase shift between sensor locations is minimal. However, in this study we show that this design guideline is not valid in the case where obstructions (such as the driver of the vehicle) are present. This study illuminates the interaction among modeling and hardware issues within the context of a real-world application. At the same time, it provides a case study illustrating RCAC and its performance and requirements in practice.

Combined state and parameter estimation and identifiability of state space realizations

Abstract: The objective of combined state and parameter estimation is to estimate both unmeasured states and unknown entries of the dynamics matrix. Since the dynamics involve products of states and parameters, this is a nonlinear estimation problem. The classical approach to this problem is to use the extended Kalman filter, although more recent techniques, such as the unscented Kalman filter, can be used. The goal of this paper is to determine conditions under which the combined state and parameter estimation problem is feasible. To do this, we recast this problem as an identifiability problem and, for several special cases, we develop necessary and sufficient conditions for identifiability, which provides necessary and sufficient conditions for feasibility of the combined state and parameter estimation problem.

Kalman-filter-based time-varying parameter estimation via retrospective optimization of the process noise covariance

Abstract: Retrospective estimation of the process noise covariance is performed by minimizing the cumulative state-estimation error based on the innovations. This technique is applied to parameter estimation problems, where the parameters to be estimated are time-varying and thus do not fit in the classical Kalman filter noise structure. This technique is compared to the standard Kalman filter with a fixed process noise covariance as well as an innovations-based adaptive Kalman filter.

Adaptive control of mass-spring systems with unknown hysteretic contact friction

Abstract: We apply retrospective cost adaptive control (RCAC) to a command-following problem for mass-spring systems with unknown contact friction. Dahl, LuGre, and Maxwell-slip models are used to generate the friction force. We consider a single-degree-of-freedom oscillator with control force applied directly to the mass, as well as a noncolocated two-degree-of-freedom oscillator with control force applied to the secondary mass and performance given by the position of the primary mass. For harmonic command following, we show that RCAC achieves internal model control without knowledge of either the friction force or the friction model.

Adaptive control of hammerstein systems with unknown input nonlinearity and partially modeled linear dynamics

Abstract: We numerically investigate that an adaptive control law achieves internal model principle control in the presence of plant input nonlinearities. We focus on retrospective cost adaptive control (RCAC) applied to Hammerstein systems with unknown input nonlinearity and limited modeling of the linear dynamics. The goal is to determine whether the control law achieves the correct gain and phase shift for internal stability along with asymptotic command following and disturbance rejection.

Adaptive control of plants that are practically impossible to control by fixed-gain control laws

Abstract: Some LTI plants are practically impossible to control due to extremely small gain and phase margins. These plants tend to be either unstable or nonminimum phase or both. Since practical control of these plants using fixed-gain controllers is not feasible, it is of interest to determine whether adaptive control can overcome these difficulties. To investigate this question, we apply retrospective cost adaptive control (RCAC) to a collection of plants that are practically impossible to control from an LTI perspective. For each plant, we introduce a destabilizing perturbation in order to determine whether or not RCAC can re-adapt in such a way as to compensate for the loss of margin and restabilize the closed-loop system without manual retuning. Since these plants are inherently difficult to control, it is of interest to determine whether or not restabilization is possible and, if so, assess the severity of the transient response.

Retrospective cost adaptive control using concurrent controller optimization and target-model identification

Abstract: Retrospective cost adaptive control (RCAC) is a discrete-time adaptive control algorithm for stabilization, command following, and disturbance rejection. RCAC requires knowledge of the nonminimum-phase (NMP) zeros in the transfer function from the control input to the performance variable. This knowledge is embedded in the target model used to define the retrospective performance variable. Without this knowledge, RCAC has a tendency to cancel unmodeled NMP zeros. The contribution of the present paper is an extension of RCAC that alleviates the need to know the NMP zeros a priori. In particular, concurrent optimization is used to update the coefficients of the controller and target model, thus providing estimates of the unmodeled NMP zeros. Since the retrospective cost is a biquadratic function of these coefficients, an alternating convex search algorithm takes advantage of the closed-form minimizers of both quadratic cost functions. For comparison, the Matlab fminsearch routine is used to jointly optimize the controller and target model. These techniques are illustrated for SISO plants that are asymptotically stable, unstable, minimum phase, and nonminimum phase.

Nonlinear aircraft flight control using the forward propagating riccati equation

Abstract: This paper presents application of the forward-propagating Riccati equation (FPRE) control for the aircraft flight control system for command following and disturbance rejection. Unlike classical finite-horizon optimal control, where the differential Riccati equation is integrated backwards in time for a given final-time condition, FPRE control uses differential equations that are integrated forward in time. Although this technique is heuristic and guarantees neither performance nor stability, simplicity of the FPRE algorithm makes it attractive for applications for nonlinear systems defined using a state-dependent coefficient parameterization. The performance of the proposed flight control system is investigated via the numerical simulations using a nonlinear model of a fixed-wing aircraft.

Adaptive non-Bayesian state estimation

Abstract: This paper presents an adaptive retrospective cost state estimation (RCSE) algorithm that uses no knowledge of the statistics of the process and measurement noise. To demonstrate the method, we investigate three cases for comparison with the Kalman filter, namely, known dynamics with known noise statistics, uncertain dynamics with known noise statistics, and uncertain dynamics with uncertain noise statistics. For numerical illustration, we apply RCSE to a damped oscillator, a damped rigid body, and a lateral aircraft model.

Gravity-gradient-stabilized spacecraft attitude estimation using rate-gyroscope measurements

Abstract: This paper considers attitude estimation of a gravity-gradient-stabilized spacecraft. In particular, an observability analysis based on the linearized equations of motion, as well as numerical simulations, show that it is possible to estimate attitude using only a rate gyroscope. The ability to estimate the attitude using only a rate gyroscope reduces the need for additional sensing hardware such as sun sensors and horizon sensors. These savings allow for more mass, volume, and power resources to be devoted to scientific payloads, communications, and other operations, which in the context of Earth orbiting microsatellites and nanosatellites, is highly desirable.

Noncolocated adaptive attitude control of a planar two-body linkage with nonminimum-phase dynamics

Abstract: Adaptive control of a two-dimensional model of a flexible spacecraft with noncolocated sensors and actuators is achieved by output feedback using knowledge of only the system's impulse response. The model is composed of two planar rigid bodies linked by a torsional spring that emulates a multibody spacecraft with base body actuation and appendage pointing. Retrospective cost adaptive control is applied to a command-following problem. The controller uses a finite impulse response filter built from the Markov parameters of the nonminimum-phase (NMP) linearized model. Accordingly, this filter is constructed to contain an estimate of the location of the NMP zero; the filter's order is chosen to correspond to the number of time steps after which the plant's step response becomes positive.

Is It Possible to Follow Position Commands Using Velocity Measurements? [Lecture Notes]

Abstract: In a simplistic sense, position can be determined from measurements of velocity by integration, and velocity can be determined from measurements of position by differentiation. In the former case, integration requires an initial condition and thus requires at least one position measurement. However, even in the case where the initial position is known, constant-but-unknown velocity-measurement noise, that is, bias, produces a spurious ramp in the computed position. On the other side of the coin, the accuracy of differentiation is limited by sensor noise, and thus approximate differentiation must be used in practice. Hence, low-frequency noise is the bane of integration, whereas high-frequency noise is the nemesis of differentiation.

Retrospective cost adaptive control with concurrent closed-loop identification of time-varying nonminimum-phase zeros

Abstract: We consider retrospective cost adaptive control (RCAC) of a plant whose NMP zeros are time-dependent. The goal is to estimate the NMP zeros and replicate the estimated NMP zeros in the target model used by RCAC. This problem is challenging due to the fact that the estimates of the locations of the NMP zeros must be sufficiently accurate at each instant of time so that the target model can correctly influence the controller adaptation. We use closed-loop identification to estimate the location of the NMP zeros. In order to enhance the accuracy of the estimation, we inject an additional noise term in order to improve persistency of the control signal. Numerical examples show the feasibility and performance of the overall approach.

Continuous-time retrospective cost adaptive control for nonminimum-phase systems

Abstract: We present a continuous-time output-feedback direct adaptive control method to deal with command following or disturbance rejection problem for systems that are possibly nonminimum phase and exponentially unstable. The adaptive control algorithm requires knowledge of the nonminimum-phase zeros of the transfer function from the control to the output. However, the knowledge of the characteristics or measurement of the reference command and disturbance is not required. The closed-loop stability is analyzed and the convergence of tracking error is proved under assumptions. The proposed adaptive control method is an extension of discrete-time retrospective cost adaptive control to continuous-time. Interpretation of retrospective cost adaptive control is provided, which is applicable for both discrete-time and continuous-time version. Numerical examples show that the proposed adaptive control method is effective for command following problem with unmeasured reference command and, in addition, robust to errors in the nonminimum-phase zero estimates.

Adaptive error correction for discrete-time systems with amplitude and rate saturation

Abstract: We present an add-on scheme for multi-input, multi-output systems with a nominal controller in order to avoid problems caused by amplitude and rate saturation such as integrator windup and phase lag. No modelling information is needed except the number of input and output. The adaptive error correction estimates the effect of control saturation on the command following error and directly modify the command following error that is input to the nominal controller in order to prevent further saturation. A retrospective cost optimization algorithm is applied to obtain the correction on-line based on measurements of the command following error and the amount of control saturation. Different from anti-windup scheme, which compensates the control command to enlarge the convergence region, this scheme intends to adaptively contain the error inside the convergence region provided by the nominal controller. Numerical examples show that, together with fixed-gain proportional-integral type controller, the adaptive error correction scheme can prevent integrator windup and phase lag for asymptotically stable plant and critically stable plant in the presence of amplitude and rate saturation.

Adaptive decentralized control with nonminimum-phase closed-loop channel zeros

Abstract: We apply retrospective cost adaptive control (RCAC) to a two-channel decentralized disturbance rejection problem. It is shown that the closed-loop channel zeros for each subcontroller consist of the plant zeros and poles of the remaining subcontroller. The nonminimum-phase (NMP) closed-loop channel zeros are included in the modeling information required by RCAC. Two adaptation schemes are presented. In one-controller-at-a-time adaptation, one subcontroller is adapted with the other subcontroller fixed at zero. The first subcontroller is then fixed while the second subcontroller is adapted taking into account the NMP closed-loop channel zeros. We also consider concurrent adaptation, where both controllers are updated at the same time. Finally, we apply this technique to decentralized control of the position and shape of a 2DOF lumped flexible body.