State variable feedback design

Consider a system described by the state and output equations 

In equation (8.94) the matrix (A - BK) is the closed-loop system matrix. 

For the system described by equation (8.92), and using equation (8.52), the characteristic equation is given by 

The roots of equation (8.96) are the closed-loop poles or eigenvalues. 

---

The pole placement design predicates on the feedback of all the state variables x (Fig. 9.1). Under many circumstances, this may not be true. We have to estimate unmeasureable state variables or signals that are too noisy to be measured accurately. One approach to work around this problem is to estimate the state vector with a model. The algorithm that performs this estimation is called the state observer or the state estimator. The estimated state X is then used as the feedback signal in a control system (Fig. 9.3). A full-order state observer estimates all the states even when some of them are measured. A reduced-order observer does the smart thing and skip these measurable states. 

If all the state variables are not measured, an observer should be implemented. In the Figure 14, the jacket temperature is assumed as not measured, but it can be easily estimated by the rest of inputs and outputs and based on the separation principle, the observer and the control can be calculated independently. In this structure, the observer block will provide the missing output, the integrators block will integrate the concentration and temperature errors and, these three variables, together with the directly measured, will input the state feedback (static) control law, K. Details about the design of these blocks can be found in the cited references. 

Block Diagram Analysis One shortcoming of this feedforward design procedure is that it is based on the steady-state characteristics of the process and, as such, neglects process (Ramies (i.e., how fast the controlled variable responds to changes in the load and manipulated variables). Thus, it is often necessary to include dynamic compensation in the feedforward controller. The most direct method of designing the FF dynamic compensator is to use a block dir rram of a general process, as shown in Fig. 8-34, where G, represents the disturbance transmitter, (iis the feedforward controller, Cj relates the disturbance to the controlled variable, G is the valve, Gp is the process, G is the output transmitter, and G is the feedback controller. All blocks correspond to transfer fimetions (via Laplace transforms). 

Chemical conversion is an effective way to counteract the accumulation of impurities due to positive feedback. Also, changing the connectivity of units may be used to modify the effect of interactions, for example by preventing an excessive increase in recycles due to snowball effects. Effective plantwide control structures may imply controlled and manipulated variables belonging to different but dynamically neighbouring units. The methodology to evaluate the dynamic inventory of impurities consists of a combination of steady state and dynamic flowsheeting with controllability analysis. This is used to assess the best flowsheet alternative and propose subsequent design modifications of units. Case Study 3 in Chapter 17 will present this problem in more detail. 

The use of an unknown value can assist in protecting against this type of design flaw. At startup and after temporary shutdown, process variables that reflect the state of the controlled process should be initialized with the value unknown and updated when new feedback arrives. This procedure will result in resynchronizing the process model and the controlled process state. The control algorithm must also account, of course, for the proper behavior in case it needs to use a process model variable that has the unknown value. 

Attractive in its simplicity, yet complex in its behavior, the Continuous Stirred Tank Reactor has, for the better part of a century, presented the research community with a rich paradigm for nonlinear dynamics and complexity. The root of complex behavior in this system stems from the combination of its open system feature of maintaining a state far from equilibrium and the nonlinear non-monotonic feedback of various variables on the rate of reaction. Its behavior has been studied under various designs, chemistries and configurations and has exhibited almost every known nonlinear dynamics phenomenon. The polymerization chemistry has especially proven fruitful as concerns complex dynamics in a CSTR, as attested to by the numerous studies reviewed in this chapter. All indications are that this simple paradigm will continue to surprise us with many more complex discoveries to come. 

The state feedback controller is designed using Linear Quadratic Regulator (LQR) optimal design to minimize the performance variables in s without using excessive control input. There are two controller options. The first one minimizes the Kqz deviation from desired while the second minimizes deviations of both Kqz and net power. The goal for each is formulated with a suitable cost function. The cost fimction for the first is given. 

---