State-Space Representation

For coupled simulation of a dynamic system with second order differential equations together with its control part, the transformation to a set of first order of differential equations into the so called state space representation is desirable in order to simplify the solution process. This has to be achieved by a duplication of the number of equations. 

The state-space representation of a linear or linearized system consists of the system equation 

The system s dynamic response variables such as displacements and velocities are contained in the state vector x n x 1). Physical quantities that exert excitations on the system (e. g. external forces and actuator forces) are collected in an input vector u p x 1), and measured quantities (sensor signals) in an output vector y(g x 1). For actively controlled adaptronic systems, the task is to generate a suitable input u t) from a given output y t) such that the system exhibits desirable dynamic behaviour. 

The matrix A n x n) is called the state or system matrix, which comprises the properties of the adaptronic (controlled) plant. The input matrix B n X p) maps the excitation and control forces to the relevant degrees of freedom of the plant model, while the output matrix C q x n) relates the state vector with measured responses. The feed through matrix D q x n) 

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If Pmfv) and the plant uncertainty A(.v) are combined to give P(.v), then Figure 9.29 can be simplified as shown in Figure 9.30, also referred to as the two-port state-space representation. 

This converts a transfer function into its state-space representation using tf2ss(num, den) and back again using ss2tf (A,B, C, D, iu) when iu is the itii input u, normaiiy i. 

Example 8.4 transfer function to state space representation... 

The way we have stated the domain theory for the state-space representation has enabled us to avoid making explicit reference to the alphabet symbol properties. However, if in other formulations we need to refer to these properties, we would again use a recursive parsing of the list of symbols to enable generalization over the size of the alphabet. 

Ibaraki, T., Branch and bound procedure and state-space representation of combinatorial optimization problems. Inf. Control 36,1-27 (1978). 

We do not need to carry the algebra further. The points that we want to make are clear. First, even the first vessel has a second order transfer function it arises from the interaction with the second tank. Second, if we expand Eq. (3-46), we should see that the interaction introduces an extra term in the characteristic polynomial, but the poles should remain real and negative.1 That is, the tank responses remain overdamped. Finally, we may be afraid( ) that the algebra might become hopelessly tangled with more complex models. Indeed, we d prefer to use state space representation based on Eqs. (3-41) and (3-42). After Chapters 4 and 9, you can try this problem in Homework Problem 11.39. 

Understand the how a state space representation is related to the transfer function representation. 

Example 4.1 Derive the state space representation of a second order differential equation of a form similar to Eq. (3-16) on page 3-5 ... 

X Example 4.3 Let s try another model with a slightly more complex input. Derive the state space representation of the differential equation... 

Example 4.5 Derive the state space representation of two continuous flow stirred-tank reactors in series (CSTR-in-series). Chemical reaction is first order in both reactors. The reactor volumes are fixed, but the volumetric flow rate and inlet concentration are functions of time. 

We use this example to illustrate how state space representation can handle complex models. First, we make use of the solution to Review Problem 2 in Chapter 3 (p. 3-18) and write the mass balances of reactant A in chemical reactors 1 and 2 ... 

To derive the state space representation, one visual approach is to identify locations in the block diagram where we can assign state variables and write out the individual transfer functions. In this example, we have chosen to use (Fig. E4.6)... 

An important reminder Eq. (E4-23) has zero initial conditions x(0) = 0. This is a direct consequence of deriving the state space representation from transfer functions. Otherwise, Eq. (4-1) is not subjected to this restriction. 

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... 

The point is that state space representation is general and is not restricted to problems with zero initial conditions. When Eq. (4-1) is homogeneous (/.< ., Bu = 0), the solution is simply... 

While there is no unique state space representation of a system, there are standard ones that control techniques make use of. Given any state equations (and if some conditions are met), it is possible to convert them to these standard forms. We cover in this subsection a couple of important canonical forms. 

A tool that we should be familiar with from introductory linear algebra is similarity transform, which allows us to transform a matrix into another one but which retains the same eigenvalues. If a state x and another x are related via a so-called similarity transformation, the state space representations constmcted with x and x are considered to be equivalent.1... 

That includes transforming a given system to the controllable canonical form. We can say that state space representations are unique up to a similarity transform. As for transfer functions, we can say that they are unique up to scaling in the coefficients in the numerator and denominator. However, the derivation of canonical transforms requires material from Chapter 9 and is not crucial for the discussion here. These details are provided on our Web Support. 

We now finally launch into the material on controllers. State space representation is more abstract and it helps to understand controllers in the classical sense first. We will come back to state space controller design later. Our introduction stays with the basics. Our primary focus is to learn how to design and tune a classical PID controller. Before that, we first need to know how to set up a problem and derive the closed-loop characteristic equation. 

We now return to the use of state space representation that was introduced in Chapter 4. As you may have guessed, we want to design control systems based on state space analysis. State feedback controller is very different from the classical PID controller. Our treatment remains introductory, and we will stay with linear or linearized SISO systems. Nevertheless, the topics here should enlighten( ) us as to what modem control is all about. 

When we used root locus for controller design in Chapter 7, we chose a dominant pole (or a conjugate pair if complex). With state space representation, we have the mathematical tool to choose all the closed-loop poles. To begin, we restate the state space model in Eqs. (4-1) and (4-2) ... 

The important message is that there is no unique state space representation, but all model matrixes should have the same eigenvalues. In addition, the number of state variables is the same as the order of the process or system. 

The step () function also accepts state space representation, and to generate the unit step response is no more difficult than using a transfer function ... 

Sure enough, the results are identical. We d be in big trouble if it were not In fact, we should get the identical result with other state space representations of the model. (You may try this yourself with the other set of a,b,c,d returned by tf 2ss () when we first went through Example 4.1.)... 

And we can easily obtain a state-space representation and see that the eigenvalues of the state matrix are identical to the closed-loop poles ... 

For effective control of crystallizers, multivariable controllers are required. In order to design such controllers, a model in state space representation is required. Therefore the population balance has to be transformed into a set of ordinary differential equations. Two transformation methods were reported in the literature. However, the first method is limited to MSNPR crystallizers with simple size dependent growth rate kinetics whereas the other method results in very high orders of the state space model which causes problems in the control system design. Therefore system identification, which can also be applied directly on experimental data without the intermediate step of calculating the kinetic parameters, is proposed. 

Most design methods for multivariable controllers require a dynamic model of the process in the linear state space representation. Such a model Is given by... 

This results In a set of first-order ordinary differential equations for the dynamics of the moments. However, the population balance Is still required In the model to determine the three Integrals and no state space representation can be formed. Only for simple MSMPR (Mixed Suspension Mixed Product Removal) crystallizers with simple crystal growth behaviour, the population balance Is redundant In the model. For MSMPR crystallizers, Q =0 and hp L)=l, thus ... 

Finite difference Has simple construction Is easily extended to multiple dimensions Solution is stable for sharp gradients or in response to a concentration or temperature front Leads directly to state-space representation Is often computationally prohibitive since accurate solutions may require a large number of grid points... 

Orthogonal collocation Leads directly to state-space representation specified over the domain May lead to control modeling difficulties since inputs affect all states immediately. 

In the development of a general state-space representation of the reactor, all possible control and expected disturbance variables need to be identified. In the following analysis, we will treat the control and disturbance variables identically to develop a model of the form... 

This basic approach is really divided into several distinct categories. Two of these, Davison s method and Marshall s method, provide suitable modal reduction for the state-space representation of the methanation reactor to a 12th-order model. Comparisons of the models and discussion of additional model reduction are presented in the next section. 

Table VII shows the system eigenvalues for the full 30th-order linear state-space representation for type II conditions. As shown, the eigenvalues can be grouped into five distinct groups based on the real parts of the eigenvalues ...

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