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A method is developed for improving the stability of linear multivariable systems using output feedback. The technique, which utilizes a gradient approach, has been mechanized in a digital computer program. Illustrative results are given for a seven-state two-feedback model of the Saturn V booster.
An iterative procedure for determining the constant gain matrix that will stabilize a linear constant multivariable system using output feedback is described. The use of this procedure avoids the transformation of variables which is required in other procedures. For the case in which the product of the output and input vector dimensions is greater than the number of states of the plant, general solution is given. In the case in which the states exceed the product of input and output vector dimensions, a least square solution which may not be stable in all cases is presented. The results are illustrated with examples.
Optimal constant output feedback gains for linear multivariable systems, noting control vector as time invariant function of output vector
This paper considers the l1-optimal control problem for general rational plants. It is shown that, for plants with no poles or zeros on the unit circle, an optimal compensator exists and that the resulting closed loop transfer function is polynomial whenever there are at least as many controls as regulated outputs and at least as many measurements as exogeneous inputs. Exactly or approximately, optimal rational compensators can be obtained by solving a sequence of finite linear programs for the coefficients of a polynomial closed-loop transfer function. No assumptions on plant poles or zeros are required to obtain at least approximately optimal compensators. It is shown that constrained problems in which a set of outputs is regulated subject to l(infinity)-norm constraints on another set of outputs can be solved using a slight modification of the same algorithm.
A minimax design method is applied to the problem of obtaining an acceptable output feedback matrix for linear multivariable systems with parameter uncertainty. The result is a set of nonlinear matrix equations (similar to those obtained by Levine and Athans (1970)), which must be solved for the feedback matrix. An example illustrates the technique and the fact that better results are achieved for large parameter variation than with a purely nominal design.
A multivariate statistical clustering technique--based on the k-means algorithm of Hartigan has been used to extract patterns of climatological significance from 200 years of general circulation model (GCM) output. Originally developed and implemented on a Beowulf-style parallel computer constructed by Hoffman and Hargrove from surplus commodity desktop PCs, the high performance parallel clustering algorithm was previously applied to the derivation of ecoregions from map stacks of 9 and 25 geophysical conditions or variables for the conterminous U.S. at a resolution of 1 sq km. Now applied both across space and through time, the clustering technique yields temporally-varying climate regimes predicted by transient runs of the Parallel Climate Model (PCM). Using a business-as-usual (BAU) scenario and clustering four fields of significance to the global water cycle (surface temperature, precipitation, soil moisture, and snow depth) from 1871 through 2098, the authors' analysis shows an increase in spatial area occupied by the cluster or climate regime which typifies desert regions (i.e., an increase in desertification) and a decrease in the spatial area occupied by the climate regime typifying winter-time high latitude perma-frost regions. The patterns of cluster changes have been analyzed to understand the predicted variability in the water cycle on global and continental scales. In addition, representative climate regimes were determined by taking three 10-year averages of the fields 100 years apart for northern hemisphere winter (December, January, and February) and summer (June, July, and August). The result is global maps of typical seasonal climate regimes for 100 years in the past, for the present, and for 100 years into the future. Using three-dimensional data or phase space representations of these climate regimes (i.e., the cluster centroids), the authors demonstrate the portion of this phase space occupied by the land surface at all points in space and time. Any single spot on the globe will exist in one of these climate regimes at any single point in time. By incrementing time, that same spot will trace out a trajectory or orbit between and among these climate regimes (or atmospheric states) in phase (or state) space. When a geographic region enters a state it never previously visited, a climatic change is said to have occurred. Tracing out the entire trajectory of a single spot on the globe yields a 'manifold' in state space representing the shape of its predicted climate occupancy. This sort of analysis enables a researcher to more easily grasp the multivariate behavior of the climate system.
A control system design approach for optimal stability augmentation, systems, using limited state feedback theory with the specific inclusion of the human pilot in the loop is presented. The methodology is especially suitable for application to flight vehicles exhibiting nonconventional dynamic characteristics and for which quantitative handling qualities specifications are not available. The design is based on a correlation between pilot ratings and objective function of the optimal control model of the human pilot. Simultaneous optimization for augmentation and pilot gains are required.
A unified identification and control scheme capable of achieving space at form performance objectives under nominal or failure conditions is described. Preliminary results are also presented, showing that the methodology offers much promise for effective robust control of large space structures. The control method is a multivariable, adaptive, output predictive controller called Model Predictive Control (MPC). MPC uses a state space model and input reference trajectories of set or tracking points to adaptively generate optimum commands. For a fixed model, MPC processes commands with great efficiency, and is also highly robust. A key feature of MPC is its ability to control either nonminimum phase or open loop unstable systems. As an output controller, MPC does not explicitly require full state feedback, as do most multivariable (e.g., Linear Quadratic) methods. Its features are very useful in LSS operations, as they allow non-collocated actuators and sensors. The identification scheme is based on canonical variate analysis (CVA) of input and output data. The CVA technique is particularly suited for the measurement and identification of structural dynamic processes - that is, unsteady transient or dynamically interacting processes such as between aerodynamics and structural deformation - from short, noisy data. CVA is structured so that the identification can be done in real or near real time, using computationally stable algorithms. Modeling LSS dynamics in 1-g laboratories has always been a major impediment not only to understanding their behavior in orbit, but also to controlling it. In cases where the theoretical model is not confirmed, current methods provide few clues concerning additional dynamical relationships that are not included in the theoretical models. CVA needs no a priori model data, or structure; all statistically significant dynamical states are determined using natural, entropy-based methods. Heretofore, a major limitation in applying adaptive controllers to LSS problems has been the requirement for accurate models. We feel that MPC coupled with the CVA identification scheme will resolve this limitation. The results are based on research performed under an SBIR grant, administered by the Air Force Office of Scientific Research.
This paper presents the design of digital rebalance loops for a tuned-rotor gyro which is intended to be an attitude sensor in a strapdown inertial measurement unit. A tuned-rotor gyro is a two-degree-of-freedom gyro which has two input axes, with cross-coupling between them. The rebalance loop of the gyro serves for two purposes. First, it restores the position of the rotor to its null position after experiencing an attitude input. Second, the rebalancing signal, which is proportional to the attitude change, is calibrated to give the desired attitude information. Since the gyro has more than one input and one output, a multivariable control technique is needed for designing the rebalance loop. The analytic model for the gyro is described and the available design methods are examined. A new root-locus design technique specially developed for the present problem is outlined. The design of the rebalance loop using the new technique is given. The computer simulation result of the designed system is presented and discussed.
Control system adapts to changing design requirements and operating conditions. Proposed control system for gas/tungsten arc welding requires only that operator specifies such direct parameters of welds as widths and depths of penetration. In control system for robotic welder, components and functions intimately connected with welding process assigned to controller domain. More general functions assigned to supervisor domain. Initial estimate of indirect parameters of welding process applied to system only at beginning of weld (t=0); after start of welding, outputs from multivariable controller takes place of estimate.
A multivariable control design procedure based on output feedback regulator (OFR) theory is applied to the F100 turbofan engine. Results for the OFR design are compared to a design based on linear quadratic regulator (LQR) theory. This LQR design was obtained as part of the F100 Multivariable Control Synthesis (MVCS) program. In the MVCS program the LQR feedback control was designed in a reduced dimension state space and then applied to the original system. However, the OFR feedback control is designed in the full order state space and thus eliminates any need for model reduction techniques. Using the performance measure and control structure of the MVCS program LQR design, an equivalent OFR feedback control is obtained. The flexibility of the OFR as a control design procedure is demonstrated and differing feedback control structures are evaluated.
A multivariable control design procedure based on output feedback regulator (OFR) theory is applied to the F100 turbofan engine. Results for the OFR design are compared to a design based on linear quadratic regulator (LQR) theory. The OFR feedback control is designed in the full order state space and thus eliminates any need for model reduction techniques. Using the performance measure and control structure of the LQR design, an equivalent OFR feedback control is obtained. The flexibility of the OFR as a control design procedure is demonstrated, and differing feedback control structures are evaluated.
The problem of pole-assignment in a linear time-invariant multivariable system using output feedback is considered. New sufficient conditions are derived to assign an almost arbitrary set of min (n, m+r-1) distinct eigenvalues, where n, m, and r are the number of states, inputs, and outputs, respectively. The analysis also highlights the freedom in selection of closed-loop eigenvectors which can be used for response shaping.
The present treatment of the related problems of minimal inversion and perfect output control in linear multivariable systems uses a simple analytical expression for the inverse of a square multivariate system's transfer-function matrix to construct a minimal-order inverse of the system. Because the poles of the minimal-order inverse are the transmission zeros of the system, necessary and sufficient conditions for the inverse system's stability are simply stated in terms of the zero polynomial of the original system. A necessary and sufficient condition for the existence of the required controllers is that the plant zero polynomial be neither identical to zero nor unstable.
Given a linear time invariant multivariable system with m inputs and p outputs, it was shown that p closed loop poles of the system can be preassigned arbitrarily using constant gain output feedback provided (A circumflex, B circumflex) is controllable. These data show that if (A circumflex, B circumflex, C circumflex) is controllable and observable, and Rank B circumflex = m, Rank C circumflex = p, then max (m,p) poles of the system can be assigned arbitarily using constant gain output feedback. Further, it is shown that in some cases more than max (m,p) poles can be arbitrarily assigned. A least square design technique is outlined to approximate the desired pole locations when it is not possible to place all the poles.
A recently developed methodology of adaptive dual control based upon sensitivity functions is applied here to a multivariable input-output model. The plant has constant but unknown parameters. It represents a simplified linear version of the relationship between the vibration output and the higher harmonic control input for a helicopter. The cautious and the new dual controller are examined. In many instances, the cautious controller is seen to turn off. The new dual controller modifies the cautious control design by numerator and denominator correction terms which depend upon the sensitivity functions of the expected future cost and avoids the turn-off and burst phenomena. Monte Carlo simulations and statistical tests of significance indicate the superiority of the dual controller over the cautious and the heuristic certainity equivalence controllers.
A multivariable control design procedure based on the output feedback regulator formulation is described and applied to turbofan engine model. Full order model dynamics, were incorporated in the example design. The effect of actuator dynamics on closed loop performance was investigaged. Also, the importance of turbine inlet temperature as an element of the dynamic feedback was studied. Step responses were given to indicate the improvement in system performance with this control. Calculation times for all experiments are given in CPU seconds for comparison purposes.
A multivariable control design procedure based on the output feedback regulator formulation is described and applied to an F100 turbofan engine model. Full order model dynamics, are incorporated in the example design. The effect of actuator dynamics on closed loop performance is investigated. Also, the importance of turbine inlet temperature as an element of the dynamic feedback is studied. Step responses are given to indicate the improvement in system performance with this control. Calculation times for all experiments are given in CPU seconds for comparison purposes.