Controllability and observability

The concepts of controllability and observability were introduced by Kalman (1960) and play an important role in the control of multivariable systems. 

A system is said to be controllable if a control vector u(t) exists that will transfer the system from any initial state x(to) to some final state t) in a finite time interval. 

A system is said to be observable if at time to, the system state x(to) can be exactly determined from observation of the output y(t) over a finite time interval. 

The system described by equations (8.87) is completely observable if the x matrix 

State-space methods for control system design 249 

The efficiency and proper positioning of actuators and sensors in adaptronic systems can be analysed using the concepts of controllabihty and observability. To make the basic ideas more clear, adaptronic structures are taken as an example. Loosely speaking, controllability and observabihty also mean that the actuator force and sensor vectors are not orthogonal and preferably parallel to the relevant vector (e. g. natural mode) or state to be controlled or observed. 

The dynamic behaviour of structural systems can be characterised in terms of natural frequencies and modes, including possible rigid-body modes in multi-body systems. If the natural modes of a system are supposed to be actively controlled using actuators and sensors, these elements must be able to influence and sense, respectively, the appropriate modal oscillations. If a mode cannot be detected by a given sensor, it is not observable. Analogously, a pin force actuator located in a node of a mode shape is imable to excite this mode, which is then said to be not controllable. 

If an adaptronic system is modelled in a state-space description (5.3), (5.4), its observability and controllability can be determined numerically by various methods. A common way is to compute the eigenvalues of the controllability and observability Gramians 

P and Q possess real non-negative eigenvalues. Large eigenvalues indicate good controllability and observability, respectively, while very small or zero eigenvalues correspond to non-controllable and non-observable states, respectively. Every linear time-invariant system ((5.3), (5.4)) can be transformed into its balanced realisation [4]. For collocated actuators and sensors P equals Q, with the Hankel singular values 

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This tutorial looks at how MATLAB commands are used to convert transfer functions into state-space vector matrix representation, and back again. The discrete-time response of a multivariable system is undertaken. Also the controllability and observability of multivariable systems is considered, together with pole placement design techniques for both controllers and observers. The problems in Chapter 8 are used as design examples. 

Most of our results have been obtained by TEM studies of individual tubes that can be considered nanolaboratories the ability to control and observe such small objects is very impressive. However, it also strains a limitation concerning the lack of an efficient method to generate macroscopic quantities of filled tubes, where we could be able to apply conventional macroscopic characterisation techniques. 

Before we formulate a state space system, we need to raise two important questions. One is whether the choice of inputs (the manipulated variables) may lead to changes in the states, and the second is whether we can evaluate all the states based on the observed output. These are what we call the controllability and observability problems. 

Example 4.7B Let us revisit the two CSTR-in-series problem in Example 4.7 (p. 4-5). Use the inlet concentration as the input variable and check that the system is controllable and observable. Find the state feedback gain such that the reactor system is very slightly underdamped with a damping ratio of 0.8, which is equivalent to about a 1.5% overshoot. 

Both the controllability and observability matrices are of rank two. Hence the system is controllable and observable. 

The remaining task lies in the determination of the control matrix X and observer matrix Z such that the sufficient condition for robust performance, Eq. (22.28), holds. A Lyapunov-based approach is employed to obtain these two matrices. After some lengthy and complicated manipulations of Eq. (22.29) and the control structure shown in Fig. 22.3, the following two Riccati equations are derived, whose positive-definite solutions correspond to the control and observer matrices, X and Z. 

C. Time-Domain Control and Observation of Molecular Vi brations in cars Microspectroscopy... 

Determine the so-called controllability and observability gramians, the matrices P and Q. These are solutions to the equations ... 

Stephanopoulos and Morari (1976, 1977) developed ideas related to those of Govind and Powers. They developed a modified view to selecting the manipulated and measured variables, aided by the concepts of "structural controllability" and "structural observability." Structural controllability and structural observability are similar to controllability and observability except they are based only on the zero/nonzero pattern of the appropriate matrices. The question asked is if any numbers were to be allowed in these matrices, would the system be controllable or observable. Their other heuristic arguments for accepting or rejecting alternatives are very similar to those of Govind and Powers. 

The example analyzed here is one of the simplest problems because it is two-dimensional with respect to the vectors state. The example illustrates the Kalman tracking for a system model, which is controllable and observable. To this aim, we use the following system model and prior statistics ... 

If suspicions eire aroused then case-control and observational cohort studies will be initiated. 

Moran and coworkers27 29 describe a procedure for control structure development based on the multilayer-multiechelon approach of hierarchical control theory, which is flee of heuristics. The key to the procedure is the effective application of decomposition to produce menaganble subeyslems of the problem. Examples are provided to show how structural controllability and observability can lead to a control system that is consistent with processing objectives. 

Control and observation of electrochemical and photoelectrochemical reactions at semiconductor electrodes are very important in establishing the electro-chemical/photoelectrochemical etching processes and stable photoelectrochemical cells... 

Hashemi and Epstein (1982) linearized the set of ordinary differential equations (ODEs) resulting from the application of the method of moments on an MSMPR crystallizer model and used singular value decomposition to define controllability and observability indices. These indices aid in selecting measurements and manipulated and control variables. Myerson et al. (1987) suggested the manipulation of the feed flow rate and the crystallizer temperature according to a nonlinear optimal stochastic control scheme with a nonlinear Kalman filter for state estimation. 

The whole system can be controlled and observed at a control center removed beyond a protective barrier about the nitration room. Free nitroglycerine occurs only in the separator, and of the 1,350 lb of nitroglycerine in the nitrating room (in a unit having an output of 2,500 lb per hr), only 125 lb occurs free, the remainder being in the comparatively safe emulsions. 

A linear system is said to be controllable if the system can be taken to any desired state x by controlling the input function. A linear system is said to be observable if the states x(t) can be determined from the observation of the output function. The following is a useful result for testing controllability and observability. 

The following result establishes the connection between the minimality of a system and its controllability and observability. 

Another related problem that seems worth investigation is, how the present result changes in the case of compartmental systems if one considers, instead of the core defined here, the controllable and observable part as defined in linear system theory, see, for example, Brockett (1970). 

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