Controller transfer function matrix

Of course, the matrices must have the same order, i.e., they must have the same number of rows and columns. Most of the matrices used in this book will be square (same number of rows and columns), and they wilt usually be the same order as system (the number of inputs and outputs). If there are JV manipulated, variables and N controlled variables, the process is an th order system and the process transfer function matrix will be an x JV square matrix. 

Equation (15.64) gives the effects of setpoint and load changes on tbe controlled variables in the closedloop multivariable environment. The matrix (of order N X N) multiplying the vector of setpoints is the closedloop servo transfer function matrix. The matrix (N x 1) multiplying the load disturbance is the closed-loop regulator transfer function vector. 

Exarngde 15.15. Determine the dosedloop characteristic equation for the system whose openloop transfer function matrix was derived in Example 15.14. Use a diagonal controller structure (two SI SO.controllers) that are proportional only. 

A fairly useful stability analysis method is the Niederlinski index. It can be used to eliminate unworkable pairings of variables at an early stage in the design. The settings of the controllers do not have to be known, but it applies only when integral action is used in all loops. It uses only the steadystate gains of the process transfer function matrix. 

Derive the openloop plant transfer function matrix relating controlled variables Xi and manipulated variables rtlj. 

The basic idea in multivariable IMC is the same as in single-loop IMC. The ideal controller would be the inverse of the plant transfer function matrix. This would give perfect control. However, the inverse of the plant transfer function matrix is not physically realizable because of deadtimes, higher-order denominators than numerators, and RHP zeros (which would give an openloop unstable controller). 

A distillation column has the following openloop transfer function matrix relating controlled variables (x, and Xg) to manipulated variables (reflux ratio RR and... 

The application of the SVD technique provides a measure of the controllability properties of a given d mamic system. More than a quantitative measure, SVD should provide a suitable basis for the comparison of the theoretical control properties among the thermally coupled sequences under consideration. To prepare the information needed for such test, each of the product streams of each of the thermally coupled systems was disturbed with a step change in product composition and the corresponding d3mamic responses were obtained. A transfer function matrix relating the product compositions to the intended manipulated variables was then constructed for each case. The transfer function matrix can be subjected to SVD ... 

Known results from control theory give the transfer function matrix for the companion model in terms of the model s matrices, Ji, Jx and C, as ... 

Bode diagram, 330-31, 334-37 frequency response, 323-24 interacting capacities, 197-200 noninteracting capacities, 194-96 pulse transfer function, 619 Multiple-input multiple-output system, 20 discrete-time model, 586 discrete transfer function, 612 input-output model, 83-85, 163-68 linearization, 121-26 transfer-function matrix, 164, 166 Multiple loop control systems, 394-409 Multiplexer, 560, 564 Multivariable control systems, 461-62 alternative configurations, 467-84 decoupling of loops, 503-8 design questions, 461-62 interaction of loops, 487-94 selection of loops, 494-503 Multivariable process (see Multiple-input multiple-output system)... 

Again, poles and zeros are important for evaluating stability and controllability properties of the physical system. To find the poles of an open-loop MIMO system one can use the transfer function matrix or the state-space description. They are related by ... 

The states of a dynamic system are simply the variables that appear in the time differential. The time-domain differential equation description of multivariable systems can be used instead of Laplace-domain transfer functions. Naturally, the two are related, and we derive these relationships below. State variables are very popular in electrical and mechanical engineering control problems, which tend to be of lower order (fewer differential equations) than chemical engineering control problems. Transfer function representation is more useful in practical process control problems because the matrices are of lower order than would be required by a state variable representation. For example, a distillation column can be represented by a 2X2 transfer function matrix. The number of state variables of the column might be 200. 

Comparing this with Eq. (12.13) and considering the case where the controlled variables Y are the same as the state variables, we see how the transfer function matrix Gm(s) and transfer function vector G s) are related to the A and. 6 matrices and to the D vector. 

Morari [2] identified the relationship between the invertibility of transfer function matrix of a system and its ability to move fast and smoothly from one operating condition to another and to deal effectively with disturbances. Therefore, plants that are easier to invert should be easier to control and possess greater resilience. Four fundamental factors were identified preventing the inversion of the process right-half-plane zeros, time delays, constraints on the variables, and model uncertainty. The limitations imposed by these factors on the operability of multivarible systems were explored by Morari, coworkers, and others in a series of papers [24-28]. 

A transfer function matrix, or, equivalently, the set of individual transfer functions, facilitates the design of control systems that deal with the interactions between inputs and outputs. For example, for the thermal mixing process in this section, control strategies can be devised that minimize or eliminate the effect of flow changes... 

The objective of the H2 or Hqo control algorithms is to design a controller K that minimizes the H2 or Hoo norm of the closed-loop transfer function matrix, H, from the disturbance to the output vector. By definition, the H2 norm of a stable transfer function matrix is... 

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. 

The limitation of transfer function representation becomes plain obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modem control, the method of choice is state space or state variables in time domain—essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modem control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

Qm = N X N matrix of process openloop transfer functions relating the controlled variables and the manipulated variables m = vector of N manipulated variables... 

The feedback controller matrix gives the transfer functions between the manipulated variables and the errors. 

In Chap. 15 we reviewed a tittle matrix mathematics and notation. Now that the tools are available, we will apply them in this chapter to the analysis of multivariable processes. Our primary concern is with closedloop systems. Given a process with its matrix of openloop transfer functions, we want to be able to see the effects of using various feedback controllers. Therefore we must be able to find out if the entire closedloop multivariable system is stable. And if it is stable, we want to know how stable it is. The last question considers the robustness of the controller, i.e., the tolerance of the controller to changes in parameters. If the system becomes unstable for small changes in process gains, time constants, or deadtimes, the controller is not robust. 

For the multivariable case, an uncertainty matrix A must be defined which contains the uncertainty in the various elements of the plant transfer function (and also could include uncertainties in setting control valves or in measurements). 

The use of singular value decomposition (SVD), introduced into chemical engineering by Moore and Downs Proc. JACC, paper WP-7C, 1981) can give some guidance in the question of what variables to control. They used SVD to select the best tray temperatures. SVD involves expressing the matrix of plant transfer function steady state gains as the product of three matrices a V matrix, a diagonal Z matrix, and a matrix. 

Alatiqi presented (I EC Process Design Dev. 1986, Vol. 25, p. 762) the transfer functions for a 4 X 4 multivariable complex distillation column with sidestream stripper for separating a ternary mixture into three products. There are four controlled variables purities of the three product streams (jCj, x, and Xjij) and a temperature difference AT to rninirnize energy consumptiou There are four manipulated variables reflux R, heat input to the reboiler, heat input to the stripper reboiler Qg, and flow rate of feed to the stripper Lj. The 4x4 matrix of openloop transfer functions relating controlled and manipulated variables is ... 

The decoupling structure is depicted in Figure 11. The global transfer matrix is diagonal. It is clear that the decoupling, as stated, is not always possible, as the transfer function of the different blocks should be stable and physically feasible. If it is possible, the new input variable u will control the concentration and U2 will control the temperature, and both control loops could be tuned independently. Sometimes, a static decoupling is more than enough. 

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 ... 

Early applications of MPC took place in the 1970s, mainly in industrial contexts, but only later MPC became a research topic. One of the first solid theoretic formulations of MPC is due to Richalet et al. [53], who proposed the so-called Model Predictive Heuristic Control (MPHC). MPHC uses a linear model, based on the impulse response and, in the presence of constraints, computes the process input via a heuristic iterative algorithm. In [23], the Dynamic Matrix Control (DMC) was introduced, which had a wide success in chemical process control both impulse and step models are used in DMC, while the process is described via a matrix of constant coefficients. In later formulations of DMC, constraints have been included in the optimization problem. Starting from the late 1980s, MPC algorithms using state-space models have been developed [38, 43], In parallel, Clarke et al. used transfer functions to formulate the so-called Generalized Predictive Control (GPC) [19-21] that turned out to be very popular in chemical process control. In the last two decades, a number of nonlinear MPC techniques has been developed [34,46, 57],... 

The term in braces is the process transfer function. Inasmuch as x has three components and c only two, the system is underdetermined—i.e. many combinations of x s will yield the same c s, at least in the steady-state. In this paper a method of selecting the x vector which is optimal in the steady-state is described. The method is shown schematically in Figure 1. The input vector x is determined by an uncoupling matrix Gm whose inputs come from the feedback controller Gc. A performance function is attached to the x vector. The elements of Gm are computed so as to minimize (or maximize) this function in the steady-state. 

CPM of multivariable control systems has attracted significant attention because of its industrial importance. Several methods have been proposed for performance assessment of multivariable control systems. One approach is based on the extension of minimum variance control performance bounds to multivariable control systems by computing the interactor matrix to estimate the time delay [103, 116]. The interactor matrix [103, 116] can be obtained theoretically from the transfer function via the Markov parameters or estimated from process data [114]. Once the interactor matrix is known, the multivariate extension of the performance bounds can be established. 

Let G = diag gi-) be the matrix containing the diagonal elements of G. The decentralized controller is diagonal K s) = diagik,) and each loop has the transfer function L, = g,A . 

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