Controllability linear system

R.E. Skelton. Dynamic Systems Control. Linear Systems Analysis and Synthesis. John Wiley Sons, New York, 1988. 

Zuazua E. (1995) Controllability of the linear system of thermoelasticity. J. Math. Pures Appl. 74 (4), 291-315. 

Bioprocess Control An industrial fermenter is a fairly sophisticated device with control of temperature, aeration rate, and perhaps pH, concentration of dissolved oxygen, or some nutrient concentration. There has been a strong trend to automated data collection and analysis. Analog control is stiU very common, but when a computer is available for on-line data collec tion, it makes sense to use it for control as well. More elaborate measurements are performed with research bioreactors, but each new electrode or assay adds more work, additional costs, and potential headaches. Most of the functional relationships in biotechnology are nonlinear, but this may not hinder control when bioprocess operate over a narrow range of conditions. Furthermore, process control is far advanced beyond the days when the main tools for designing control systems were intended for linear systems. 

If the state and control variables in equations (9.4) and (9.5) are squared, then the performance index become quadratic. The advantage of a quadratic performance index is that for a linear system it has a mathematical solution that yields a linear control law of the form... 

The Linear Quadratic Regulator (LQR) provides an optimal control law for a linear system with a quadratic performance index. 

When eompiling the material for the book, deeisions had to be made as to what should be ineluded, and what should not. It was deeided to plaee the emphasis on the eontrol of eontinuous and diserete-time linear systems. Treatment of nonlinear systems (other than linearization) has therefore not been ineluded and it is suggested that other works (sueh as Feedbaek Control Systems, Phillips and Harbor (2000)) be eonsulted as neeessary. 

Nonanalytic Nonlinearities.—A somewhat different kind of nonlinearity has been recognized in recent years, as the result of observations on the behavior of control systems. It was observed long ago that control systems that appear to be reasonably linear, if considered from the point of view of their differential equations, often exhibit self-excited oscillations, a fact that is at variance with the classical theory asserting that in linear systems self-excited oscillations are impossible. Thus, for instance, in the van der Pol equation... 

In 1982 the present author discovered cyclic orbital interactions in acyclic conjugation, and showed that the orbital phase continuity controls acyclic systems as well as the cyclic systems [23]. The orbital phase theory has thus far expanded and is still expanding the scope of its applications. Among some typical examples are included relative stabilities of cross vs linear polyenes and conjugated diradicals in the singlet and triplet states, spin preference of diradicals, regioselectivities, conformational stabilities, acute coordination angle in metal complexes, and so on. 

The linear property is one very important reason why we can do partial fractions and inverse transform using a look-up table. This is also how we analyze more complex, but linearized, systems. Even though a text may not state this property explicitly, we rely heavily on it in classical control. 

From the last example, we may see why the primary mathematical tools in modem control are based on linear system theories and time domain analysis. Part of the confusion in learning these more advanced techniques is that the umbilical cord to Laplace transform is not entirely severed, and we need to appreciate the link between the two approaches. On the bright side, if we can convert a state space model to transfer function form, we can still make use of classical control techniques. A couple of examples in Chapter 9 will illustrate how classical and state space techniques can work together. 

This completes our "feel good" examples. It may not be too obvious, but the hint is that linear system theory can help us analysis complex problems. We should recognize that state space representation can do everything in classical control and more, and feel at ease with the language of... 

A system is said to be completely state controllable if there exists an input u(t) which can drive the system from any given initial state xo(to=0) to any other desired state x(t). To derive the controllability criterion, let us restate the linear system and its solution from Eqs. (4-1), (4-2), and (4-10) ... 

In this study the linearized equations were used to determine the control strategy, but the non-linear equations were used to test this strategy. For small deviations from the steady state (5% or less) there is very little difference between the responses of the non-linear and the linearized system. 

Scolastico s approach toward functionalized azabicycloalkane amino acids such as 407 using an intramolecular 1,3-dipolar cycloaddition strategy is a powerful way of synthesizing the linear system with good regio- and stereo-control (Equation 110) <2005JOC4124>. 

Rusnak, I. A. Guez and I. Bar-Kana. Multiple Objective Approach to Adaptive Control of Linear Systems. In Proceedings of the American Control Conference. San Francisco, pp. 1101-1105 (1993). 

Schields, R., and Pearson, J. B. (1976). Structural controllability of multiinput linear systems. IEEE Trans. Autom. Control AC-21, 203-212. 

Almasy, G., and Sztano, T. (1975). Checking and correction of measurements on the basis of linear system model. Probl. Control Inf. Theory 4, 57. 

Jang, S. S., Josepth, B and Mukai, H. (1986). Comparison of two approaches to on-line parameter and state estimation problem of non-linear systems. Ind. Eng. Chem. Process Des. Dev. 25, 809-814. Jazwinski, A. H. (1970). Stochastic Processes and Filtering Theory. Academic Press, New York. Liebman, M. J., Edgar, T. F., and Lasdon, L. S. (1992). Efficient data reconciliation and estimation for dynamic process using non-linear programming techniques. Comput. Chem. Eng. 16, 963-986. McBrayer, K. F., and Edgar, T. F. (1995). Bias detection and estimation on dynamic data reconciliation. J Proc. Control 15, 285-289. 

Verotta, D., Concepts, properties, and applications of linear systems to describe distribution, identify input, and control endogenous substances and drugs in biological systems, Crit. Rev. Biomed. Eng., 24, 73-139, 1996. 

The addition of the feedforward controller has no effect on the closedloop stability of the system for linear systems. The denominators of the closedloop transfer functions are unchanged. 

To develop an HPLC stability-indicating method for Type I or II dissolution, the linearity must be wide enough, in combination with good sensitivity and minimal interference, to accommodate concentrations from low (possibly LOQ) to very high end, as the samples drawn represent the cumulative drug amount dissolved over time. As for an FiPLC method that is designed for Type VII dissolution, the linearity should accommodate the lower concentrations since it is a drug measurement of a controlled-release system. 

S. Rbimback. Linear Control of Systems with Actuator Constraints. PhD thesis. Division of Automatic Control, Lulea University of Technology, May 1993. 

Finally, the controller solving the robust regulation problem for linear system (1) takes the form... 

As in the case of continuous linear systems, the exponential holder will then ensure the fulfilment of the regulation conditions for a continuous linear system with a discrete controller. This result is summarized in the following theorem. 

Nonlinearity In addition, it is well known that the process kinetics shows a highly nonlinear behavior. This a serious drawback in instrumentation and automatic control because, in contrast to linear systems where the observability can be established independently of the process inputs, the nonlinear systems must accomplish with the detectability condition depending on the available on-line measurements, including process inputs in the case of non autonomous systems [23]. 

In this situation, a periodic variation of coolant flow rate into the reactor jacket, depending on the values of the amplitude and frequency, may drive to reactor to chaotic dynamics. With PI control, and taking into account that the reaction is carried out without excess of inert (see [1]), it will be shown that it the existence of a homoclinic Shilnikov orbit is possible. This orbit appears as a result of saturation of the control valve, and is responsible for the chaotic dynamics. The chaotic d3mamics is investigated by means of the eigenvalues of the linearized system, bifurcation diagram, divergence of nearby trajectories, Fourier power spectra, and Lyapunov s exponents. 

Exercise 6. Show that the equilibrium point of the model defined by Eq.(34) and the simplified model R given by Eq.(35), i.e. when the dynamics of the jacket is considered negligible, are the same. Deduce the Jacobian of the system (35) at the corresponding equilibrium point. Write a computer program to determine the eigenvalues of the linearized model R at the equilibrium point as a function of the dimensionless inlet flow 4 50. Values of the dimensionless parameters of the PI controller can be fixed at Ktd = 1-52 T2d = 5. The set point dimensionless temperature and the inlet coolant flow rate temperature are Xg = 0.0398, X40 = 0.0351 respectively. An appropriate value of dimensionless reference concentration is C g = 0.245. Does it exist some value of 2 50 for which the eigenvalues of the linearized system R at the equilibrium point are complex with zero real part Note that it is necessary to vary 2 50 from small to great values. Check the possibility to obtain similar results for the R model. 

The values of Km and T2d from Eq.(36) can be obtained from the transfer function of the linearized model at the equilibrium point, applying conventional methods from the linear control theory (see [1]). In order to investigate the self-oscillating behavior, one can determine the linearized system at the equilibrium point, and the corresponding complex eigenvalues with zero real part, when the parameters Km and of the PI controller are varied. For example, taking into account Eq.(34), the Jacobian matrix of the linearized system at dimensionless set point temperature xs is the following ... 

As with scale-up, two levels of implementation are possible. The first level only entails the ability to sense, and a directional characterization of the effect of variables. PAT methods can be extremely effective for this purpose by generating large datasets of process inputs and outputs that can then be correlated to generate statistical or polynomial control models. Provided that (i) deviations from desired set-points are small, (ii) interactions between inputs are weak, and (Hi) the response surface does not depart too much from linearity, such systems can provide the basis of an initial effort to control a system. 

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