Real-Time Numerical Differentiation of Sampled Data Using Adaptive Input and State Estimation

Real-time numerical differentiation plays a crucial role in many digital control algorithms, such as PID control, which requires numerical differentiation to implement derivative action. This paper addresses the problem of numerical differentiation for real-time implementation with minimal prior information about the signal and noise using adaptive input and state estimation. Adaptive input estimation with adaptive state estimation (AIE/ASE) is based on retrospective cost input estimation, while adaptive state estimation is based on an adaptive Kalman filter in which the input-estimation error covariance and the measurement-noise covariance are updated online. The accuracy of AIE/ASE is compared numerically to several conventional numerical differentiation methods. Finally, AIE/ASE is applied to simulated vehicle position data generated from CarSim.Comment: This paper is under review at the International Journal of Contro

Left Inversion, System Zeros, and Input Estimation

This dissertation focuses on input estimation, that is, estimation of the input to a linear system using knowledge of the output measurements and the system model for tall or square systems with full column rank. First, finite-time input estimation for discrete-time linear time-invariant (LTI) systems with zero nonzero zeros and unknown initial conditions is considered. Necessary and sufficient conditions for finite-time input estimation are derived. For systems with zero nonzero zeros, a specific construction of finite-impulse-response (FIR) delayed left inverse with minimal delay using the Smith-McMillan form at infinity is given. Since zeros play a vital role in input estimation, further research on system zeros is considered. Expressions for the number of transmission zeros and the number of infinite zeros in terms of the defect of a block-Toeplitz matrix of Markov parameters and the observability matrix are obtained. For counting zeros, these results serve as duals to the counting of poles using the block-Hankel matrix. Furthermore, the zero dynamics of input-output models are explored, and their properties are elucidated. Output zeroing in input-output models is considered and its equivalence to output zeroing in state space models is discussed. Next, retrospective cost input estimation (RCIE), which is an adaptive input estimation technique for discrete-time linear time-varying (LTV) systems that depends on a target model based on the closed-loop dynamics, is considered. In particular, the decomposition of the retrospective performance variable into the sum of a performance term and a model-matching term, which provides insight into the achievable performance of RCIE, is presented. Since the system dynamics and target model are LTV, the construction of LTV state space realizations for LTV input-output models as well as the construction of LTV input-output models for LTV state space models are given in this dissertation. Using the same technique used for RCIE, the decomposition of the retrospective performance variable in retrospective cost adaptive control (RCAC) is also derived. Finally, as an application of input estimation, causal numerical differentiation is considered. When the dynamics of the system consist of a cascade of one or more integrators, the estimates of the input provide estimates of one or more derivatives of the output signal. The performance of RCIE as a causal differentiator is analyzed through numerical simulations. RCIE as a causal differentiator is then applied to the position data of a small rover to estimate its velocity and acceleration.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/174577/1/snehasnj_1.pd

Stability analysis of switched systems with 'mixed'-negative imaginary property

This paper discusses the stability of feedback systems in which both plant and controller are switched. Switched systems considered here have all their subsystems satisfying the 'mixed'-negative imaginary property. A definition for dissipativity (for switched systems) is proposed, and dissipative switched systems are shown to be stable (under certain conditions). Switched systems with 'mixed'-negative imaginary property are shown to be dissipative and conditions for stability are derived. As an illustration of the results, a switched controller is designed for a nanopositioning stage, which has a 'mixed'-negative imaginary frequency response function. Simulations show that the closed loop is stable and the designed controller damps the resonances satisfactorily

Adaptive digital PID control of first‐order‐lag‐plus‐dead‐time dynamics with sensor, actuator, and feedback nonlinearities

Proportional‐integral‐derivative (PID) control is one of the most widely used feedback control strategies because of its ability to follow step commands and reject constant disturbances with zero asymptotic error, as well as the ease of tuning. This paper presents an adaptive digital PID controller for sampled‐data systems with sensor, actuator, and feedback nonlinearities. The linear continuous‐time dynamics are assumed to be first‐order lag with dead time (ie, delay). The plant gain is assumed to have known sign but unknown magnitude, and the dead time is assumed to be unknown. The sensor and actuator nonlinearities are assumed to be monotonic, with known trend but are otherwise unknown, and the feedback nonlinearity is assumed to be monotonic, but is otherwise unknown. A numerical investigation is presented to support a simulation‐based conjecture, which concerns closed‐loop stability and performance. Numerical examples illustrate the effect of initialization on the rate of adaptation and investigate failure modes in cases where the assumptions of the simulation‐based conjecture are violated. This paper presents an adaptive digital PID controller for sampled‐data systems with sensor, actuator, and feedback nonlinearities as well as linear continuous‐time dynamics that are first‐order lag with dead time (ie, delay). A numerical investigation is presented to support a simulation‐based conjecture on closed‐loop stability and performance. Numerical examples illustrate the effect of initialization on the rate of adaptation and investigate failure modes in cases where the assumptions of the simulation‐based conjecture are violated.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/153185/1/adc220.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/153185/2/adc220_am.pd