q-Markov Covariance Equivalent Realizations Research Articles

Q-Markov Covariance equivalent realizations for unstable and marginally stable systems

An observer-based q-Markov Covariance equivalent realization (QMC) formulation is presented in the paper. It is shown that by inserting an observer in the input–output relationship associated with the dynamical system, a QMC approach can be developed that is applicable to systems with marginally stable or unstable equilibrium points. A system of equations governing all linear state-space realizations, along with the corresponding observers, is derived from matching a set of Markov and Covariance parameters. The solution to this equation system is shown to parameterize all state-space realizations that match the pre-specified correlation functions. Four numerical examples are used to show the utility of the proposed approach. The numerical results show that the observer-based QMC presented in the paper can identify linear SISO and MIMO systems with stable, marginally stable, and unstable characteristics.

An Efficient q-Markov Covariance Equivalent Realization Approach to System Identification

An efficient q-Markov covariance equivalent realization (QMC) formulation is presented in this paper. This efficient solution matches all Markov and Covariance parameters of a finite-dimensional system using a finite amount of data. The frequency responses of that system is also matched by the efficient QMC solution. The existence condition of this efficient QMC solution and its parametrization is given in this paper. Then, a generalized algorithm for system identification using the efficient QMC is given. The example is set up by a cantilever beam. Results show that the efficient QMC fully captures Markov and covariance parameters and frequency responses of the cantilever beam.

Linear-Quadratic Tracking Control of a Commercial Vehicle Air Brake System

This article proposes to utilize linear-quadratic tracking (LQT) control to reduce the air brake system response time and vehicle stopping distance, and hence, to significantly improve the system performance of a commercial vehicle air brake system equipped with electro-pneumatic proportional valve actuators. The nonlinear dynamic model of the air brake system, consisting of a control actuator (a proportional valve) and braking actuator (the brake chamber), is developed and linearized using the q-Markov COVariance Equivalent Realization (q-Markov Cover) method in our early work. Based on the linearized dynamic model, an infinite horizon LQT controller is designed along with Kalman state estimation at each linearized operational condition. To apply the LQT control law over a wide operational range to track the target pressure, the designed controller was interpolated between the neighboring controllers to have a control law cover the entire operational range. To validate this control law, the control scheme is implemented into a dSPACE unit and validated through bench tests under different supply and reference pressures. The LQT control performance is also compared with the PID (proportional-integral-derivative) one. The bench test results confirm the effectiveness of the proposed control scheme.

Pseudo-random binary sequence closed-loop system identification error with integration control

This paper studies the closed-loop system identification (ID) error when a dynamic integral controller is used. Pseudo-random binary sequence (PRBS) q-Markov covariance equivalent realization (Cover) is used to identify the closed-loop model, and the open-loop model is obtained based upon the identified closed-loop model. Accurate open-loop models were obtained using PRBS q-Markov Cover system ID directly. For closed-loop system ID, accurate open-loop identified models were obtained with a proportional controller, but when a dynamic controller was used, low-frequency system ID error was found. This study suggests that extra caution is required when a dynamic integral controller is used for closed-loop system identification. The closed-loop identification framework also has significant effects on closed-loop identification error. Both first- and second-order examples are provided in this paper.

Q-Markov covariance equivalent realizations in fixed and floating point computations

This paper produces all linear state space models which match a prespecified set of input/output cross-correlation and output autocorrelation data, when the model is installed in a computational environment with specified bits assigned to the fixed-point or floating point simulation. With the assumed round-off noise model, these results allow the design of digital simulations or controller realizations with no error within the specified set of cross-correlation and autocorrelation data.

DATA-BASED CLOSED-LOOP SYSTEM SIMULATION

This paper provides a framework for control design and simulation of a closed-loop system with partial input/output data of a plant. Given the input/output crosscorrelation and output autocorrelation data of an open loop dynamic system, a simulation model implemented in fixed-point digital devices which matches these data is obtained using q-Markov Covariance Equivalent Realizations. These results allow the design of digital simulations with no error within the specified set of crosscorrelation and autocorrelation data. When a linear approximation of the plant is assumed, an LQG controller can be presented solely in terms of the input/output crosscorrelation data. This is the so-called the Markov data-based LQG control. With both the simulation model of the plant and the controller in hand, a closed-loop simulation can be constructed. This is yet another example showing that significant work can be done with very limited information of a plant.

Frequency Domain Structural System Identification by Observability Range Space Extraction

This paper presents the Frequency Domain Observability Range Space Extraction (FORSE) identification algorithm. FORSE is a singular value decomposition based identification algorithm which constructs a state space model directly from frequency domain data. The concept of system identification by observability range space extraction was developed by generalizing the Q-Markov Covariance Equivalent Realization and Eigensystem Realization Algorithm. The numerical properties of FORSE are well behaved when applied to multi-variable and high dimensional structural systems. It can achieve high modeling accuracy by properly overparameterizing the system. The effectiveness of this algorithm for structural system identification is demonstrated using the MIT Middeck Active Control Experiment (MACE). MACE is an active structural control experiment to be conducted in the Space Shuttle middeck. Results of ground experiments using this algorithm will be discussed.

Q-Markov Cover identification using pseudo-random binary signals

The original q-Markov COVariance Equivalent Realization (q-Markov Cover) method for identification required white noise test signals, which cannot be generated exactly. This paper replaces the unrealizable white-noise signal with a realizable signal (the pseudo-random binary signal, PRBS), and proves that when the period of the PRBS approaches infinity the q-Markov Cover algorithm, operating with PRBS, matches the first q Markov and covariance parameters (as in the original theory with white-noise test signals). The existing q-Markov Cover algorithm will fail to match covariance and Markov parameters exactly due to non-white test signals. The new algorithm will fail to match covariance and Markov parameters exactly due to the finite period of PRBSs. We demonstrate however that the results with PRBSs are far superior to results with ‘white’ noise signals. Apparently, the ‘non-whiteness’ of the test signal degrades the identification performance worse than the ‘non-infinite’ period of the PRBS.

Modeling Hubble Space Telescope flight data by Q-Markov cover identification

This Paper presents a state space model for the Hubble space telescope under the influence of unknown disturbances in orbit. This model was obtained from flight data by applying the Q-Markov Covariance Equivalent Realization identification algorithm. This state space model guarantees the match of the first Q Markov parameters and covariance parameters of the Hubble system. The flight data were partitioned into high and low frequency components for more efficient Q-Markov Cover modeling, to reduce some computational difficulties of the Q-Markov Cover algorithm. This identification revealed more than 20 lightly-damped modes within the bandwidth of the attitude control system. Comparisons with the analytical (TREETOPS) model are also included.

Integrated modeling and controller design with application to flexible structure control

It is well known that modeling and control problems are not independent. Hence, the final step in any practical control design is a ‘try and see’ procedure. There will always be a need for on-line tuning by the test engineer. The purpose of mathematical theories of modeling and control is therefore not to replace the ‘try and see’ requirement of the engineer, but to help reduce the number of iterations the engineer must use to find a suitable solution to the real physical problem (which defies exact mathematical description). This paper presents an iterative method to integrate the two problems of modeling and control and demonstrates the procedure in a physical application. Specifically, we present a controller design scheme integrating a Q-Markov Covariance equivalent realization (QMC) identification algorithm, a Modal Cost Analysis (MCA) model reduction algorithm, and an Output Variance Constraint (OVC) controller design algorithm. The identified model is used as a truth model for subsequent (off-line) controller evaluation. In the first step of the integrated procedure, this model is reduced for controller design. Closed-loop evaluation provides information to improve the reduced model (this is the integration of the two model reduction and control disciplines that is contributed by this paper). The process repeats iteratively for low order and high performance controller design. This procedure is applied to NASA's ACES flexible structure at the Marshall Space Flight Center. The experimental results demonstrate the effectiveness of the procedure for large flexible structure control.

Q-Markov covariance equivalent realization and its application to flexible structure identification

The Q-Markov covariance equivalent realization algorithm is applied to NASA's Minimast structure to identify state-space models for the purpose of designing closed-loop controllers, and laboratory test results are shown for the identification and for the closed-loop performance. The paper also presents for the first time a deterministic formulation of covariance parameters from a pulse response, a stochastic formulation of Markov parameters from a white-noise response, and a simple Q-Markov covariance equivalent realization formulation. This requires only pulse laboratory tests or only white-noise laboratory tests to generate Q-Markov covariance equivalent realizations.

A new formulation of Q-Markov covariance equivalent realization

The Q-Markov Covariance Equivalent Realization ( Q-Markov Cover) is a state space realization which matches the first Q Markov parameters and covariance parameters of a system. King, Desai, and Skelton [5] developed a generalized approach to generate Q-Markov Covers. Desai and Skelton [3] presented a recursive computational algorithm for generating Q-Markov Covers when Q is very large. Based on the works of the above two papers, we developed a new formula for generating Q-Markov Covers, and also an algorithm to compute them when Q is very large. The new formula has been applied for model identification of large flexible structures by Liu and Skelton [6–8]. Compared with previously developed formulas and algorithms, the new formula and algorithm are relatively straightforward and easy to program. This formula may simplify the further analytical study of Q-Markov Covers and its application to system identification.

Weighted q-Markov covariance equivalent realizations

Weighted q-Markov covariance equivalent realizations

Weightedq-Markov covariance equivalent realizations

Abstract Reduced order models which preserve the first q Markov parameters and the first q covariances are called q-Markov covariance equivalent realizations (q-Markov COVERs). This paper extends these notions to the case where the input is coloured instead of white noise. We call this the ‘weighted q-Markov COVER’.

A generalized approach to q-Markov covariance equivalent realizations for discrete systems

A new algorithm is developed which constructs q-Markov covariance equivalent realizations ( q-Markov COVERs) of multivariable linear discrete time systems. A q-Markov COVER is a state realization which matches exactly a finite portion of the impulse response sequence (Markov parameters) and output autocorrelation sequence (covariance parameters) of a linear system. The algorithm generates the q-Markov COVERs starting with only the basic data, namely the q-Markov parameters and the q covariance parameters. The algorithm leads to a parameterization of all minimal stable q-Markov COVERs which can be constructed. Since the data are easily computed from a given state space model, the algorithm presented will be useful for both realization and model reduction of multivariable linear discrete time systems. An example is presented to demonstrate the algorithm.

Q-Markov covariance equivalent realizations

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