# CHAPTER 4 STATE FEEDBACK AND OUTPUT FEEDBACK CONTROLLERS

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1 54 CHAPTER 4 STATE FEEDBACK AND OUTPUT FEEDBACK CONTROLLERS 4.1 INTRODUCTION In control theory, a controller is a device which monitors and affects the operational conditions of a given dynamic system. The operational conditions are referred to as output variables of the system which can be affected by adjusting certain input variable. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. Control systems are designed to perform specific tasks. The requirements imposed are usually called performance specifications. The specifications may be transient response requirement (maximum overshoot, settling time) and steady state requirements (steady state error). Figure 4.1 Block diagram for control system

2 55 In Figure 4.1, block diagram for control system is shown. The two fundamental concepts of control systems are controllability and observability. Controllability deals with the problem of whether it is possible to move a system from a given initial state to an arbitrary state. If a system is said to be controllable, it is possible to transfer the system from any initial state to any other state in a finite number of sampling periods by means of the unbounded control vector. Thus the concept of controllability is concerned with the existence of a control vector that can cause the system state to reach some arbitrary state. If any state variable is independent of the control signal, then it is impossible to control this state variable and therefore the system is uncontrollable. The solution to an optimal control problem may not exist if the system considered is not controllable. Observability deals with the problem of determining the state of a dynamic system from observations of the output and control vectors in a finite number of sampling periods. If a system is said to be observable, it is possible to determine the initial state from the observation of the output and the control vectors over a finite number of sampling periods. The concept of observability is useful in solving the problem of reconstructing unmeasured state variables. 4.2 REVIEW ON STATE FEEDBACK CONTROLLER The main design approach for systems described in state space form is the use of state feedback. In state feedback controller, the states of the system are considered. The control signal is determined by an instantaneous state. Such a scheme is called state feedback. State feedback control can be realized by two methods. 1. Pole placement technique. 2. Linear Quadratic Regulator.

3 Pole placement Technique Pole placement is a method employed in feedback control system to place the closed-loop poles of a plant in pre-determined locations. Placing poles is desirable because the location of the poles corresponds directly to the Eigen values of the system which controls the behavior of the system. Assume that all state variables are measurable and available for feedback. In this technique, if the system is completely state controllable, then the desired poles are chosen by means of state-feedback through an appropriate state feedback gain matrix. This technique ensures both transient and frequency response requirements and also the steady-state requirements. In conventional design approach, only the dominant closed loop poles are specified whereas pole-placement approach specifies all closed-loop poles. Since it specifies all closed-loop poles, all state variables can be measured or else state observer is needed to estimate the states. There are three methods to determine the required state feedback gain matrix 1. Transformation matrix method 2. Direct substitution method 3. Ackermann s method Figure 4.2 State feedback control system

4 57 The gain matrix is not unique for a given system but depends on the desired closed-loop pole locations selected, which determines the speed and damping of the response. The selection of the desired closed-loop poles or the desired characteristic equation is a compromise between the rapidity of the response of the error vector and the sensitivity to disturbances and measurement noises. A State feedback controller is shown in Figure State Observer In pole placement technique, an assumption is made that all state variables are available for feedback to design the control system. In practice, all states are not available for measurement. So the unmeasured states are estimated by designing an observer or estimator. A device that estimates or observes the states is known as state observer. r u x y K r H Z -1 C Process G L ŷ H Z -1 xˆ - C G Observer - K Figure 4.3 Observer based State feedback control system

5 58 A state observer as in Figure 4.3 estimates the state variables based on the measurement of the outputs and the control variables and it should be designed only if the observability conditions are satisfied Effects of the Addition of an Observer to State Feedback In the pole placement design process, it is assumed that the actual state is available for feedback. In practice, the actual state may not be measurable, so it is necessary to design a state observer. Therefore the design process involves a two stage process. First stage includes determination of the feedback gain matrix to yield the desired characteristic equation and the second stage involves the determination of the observer gain matrix to yield the desired observer characteristic equation. The closed loop poles of the observed-state feedback control system consist of the poles due to the poleplacement design and the poles due to the observer design. If the order of the plant is n, then the observer is also n th order and the resulting characteristic equation for the entire closed-loop system becomes the order of 2n. The desired closed-loop poles to be generated by state feedback are chosen in such a way that the system satisfies the performance requirements. The poles of the observer are usually chosen so that the observer response is much faster than system response. A rule of thumb is to choose an observer response at least two to five times faster than system response. The maximum speed of the observer is limited only by the noise and sensitivity problem involved in the control system. Since the observer poles are placed left of the desired closedloop poles in the pole placement process, the closed loop poles will dominate the response. 4.3 MULTIRATE OUTPUT FEEDBACK (MROF) If the states are available for measurement state feedback provides a simplest way of designing a controller. Some times the state feedback

6 59 becomes inevitable due to incomplete state information. In reality most of the states are observable but they are immeasurable. So it is essential to find a controller based on the system output which is always measurable. The state feedback control law requires the design of state observer i.e. the dynamic compensators. This increases the implementation cost and reduces the reliability of the control system. More over in observer based design, even slight variation of the model parameters from their nominal value may result insignificant degradation of closed loop stability. The other problem with observer based controller is that the state feedback and state estimation cannot be separated in face of uncertainty (Werner and Furuta 1995b). Assuming that a simultaneously stabilizing state feedback gain has been found, it is possible to use an algorithm to search for a simultaneously stabilizing full order observer gain, but this depends on the state feedback gain previously obtained. Instead of searching for the dynamic compensator parameters, the problem can be transformed into an equivalent static output feedback problem. Hence it is desirable to go for an output feedback design. The static output feedback problem is the most important open question in control theory (Kimura 1994). It is not appreciated for single input system as it is essential to match the number of inputs to the number of poles of the system i.e. it needs n-outputs to assign all the n-poles of the system. So the static output feedback has no real significance. As in most cases the number of outputs is less than the system order, the static output feedback would not be a correct option for single input single output system. The static output feedback is one of the most investigated problems in control theory. It is the simplest closed loop control but it will not guarantee the closed loop stability (Syrmos et al 1997). Jinhui Zhang and Yuanqing Xia (2010) states that static output feedback controller have less computational and hardware overheads than an observer based approach.

7 60 Hence the dynamic output feedback comes into picture. In dynamic output feedback, the feedback function is a transfer function rather than a constant vector. In this both, the poles and zeros of the systems are matched using a dynamic compensator. The order of the dynamic compensator would be very large. If the system has n-poles and m-zeros, the compensator would be of the order (m+n). Moreover since the emphasis here is on pole placement, the dynamic output feedback method used should place 2n-poles of the closed loop system. The dynamic output feedback involves more dynamics and complex design. External Input E Plant B x x C y A Z -1 [ ] Unit Delay State Computation Output Stacker Control Input Controller Figure 4.4 Block Diagram for Fast Output Sampling Controller In recent past, multirate output feedback controllers were applied for large scale systems and systems with incomplete state information. Multirate controllers can outperform single rate linear time invariant controllers due to their time varying nature. If the digital controller is so

8 61 designed that the control signal and sensor output are sampled at different rates, then such a control is called as multirate control (Kranc 1957). To assure the stability and performance, the multirate output feedback is introduced which also maintains the simplicity of the static output feedback. Multirate output feedback technique is different from the observer based technique in the sense that the system states are computed exactly just after one sampling interval as opposed to infinite time taken by an observer. Further the time delay required for control law implementation is avoided as present outputs or control inputs are used to compute the states. In case of multirate output feedback the error between the computed state and the actual state of the system goes to zero once a multirate sampled measurement is available, where as in observer, the error between the estimated and actual system state goes to zero only at infinite time (Datatreya Reddy et al 2007). In Multirate output feedback, the system output and the control input are sampled at a rate faster than the other and only the system outputs and past control inputs are used to compute the control input. An attractive feature of MROF controller is that they allow a simultaneous design for a family of models. Multirate output feedback can be realized using Fast Output Sampling (FOS) or by Periodic Output Feedback (POF). Block diagram for FOS controller is shown in Figure 4.4. In fast output sampling, the system output is sampled faster than the control input and vice versa in periodic output feedback. For any controllable and observable system, it is possible to realize the performance of a state feedback controller by using only the system output with multirate output feedback (Bandyopadhyay et al 2006). Unlike the static output feedback, fast output sampling feedback always guarantees the stability of the closed loop system (Sharma et al 2003).

9 62 Patient s tolerance considerations, limit the number of thermocouples that can be inserted into the body. So hyperthermia system is a system with incomplete state information, this demands the need for estimator design. Hence it is rather desirable to go for an output feedback design. The output feedback needs only the measurement of system output unlike the state feedback which requires the knowledge of the states or a state estimator. This chapter summarizes the merits and demerits of state feedback and output feedback controller.this also gives information on effect of adding an observer to a system. Finally it presents the advantages of multirate output feedback controller and justifies the need for using multirate output feedback controller for hyperthermia system.

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