Pole placement

Digital compensator design using pole placement... 

Fig. 7.25 Identical continuous and discrete step responses as a result of pole placement.

Here pole placement is used to design a digital compensator that produces exactly the step response of the continuous system. 

Example 7.8 Digital Compensator Design using Pole Placement... 

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. 

Full-order observer design using pole placement A=[0 l -2 -3]... 

Fig. A1.4 Inverted pendulum control system design using pole placement...

Pole placement design of state feedback systems. Application of the Ackermann s formula. 

This is the result of full state feedback pole-placement design. If the system is completely state controllable, we can compute the state gain vector K to meet our selection of all the closed-loop poles (eigenvalues) through the coefficients a . 

There are other methods in pole-placement design. One of them is the Ackermann s formula. The derivation of Eq. (9-21) predicates that we have put (9-13) in the controllable canonical form. Ackermann s formula only requires that the system (9-13) be completely state controllable. If so, we can evaluate the state feedback gain as 1... 

The state space state feedback gain (K2) related to the output variable C2 is the same as the proportional gain obtained with root locus. Given any set of closed-loop poles, we can find the state feedback gain of a controllable system using state-space pole placement methods. The use of root locus is not necessary, but it is a handy tool that we can take advantage of. 

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. 

We have the very important result that choices for the eigenvalues for the pole-placement design and the observer design can be made independently. Generally, we want the observer response to be two to five times faster than the system response. We should not have to worry about saturation since the entire observer is software-based, but we do have to consider noise and sensitivity problems. 

This paper extends previous studies on the control of a polystyrene reactor by including (1) a dynamic lag on the manipulated flow rate to improve dynamic decoupling, and (2) pole placement via state variable feedback to improve overall response time. Included from the previous work are optimal allocation of resources and steady state decoupling. Simulations on the non-linear reactor model show that response times can be reduced by a factor of 6 and that for step changes in desired values the dynamic decoupling is very satisfactory. 

The present paper applies state variable techniques of modern control theory to the process. The introduction of a dynamic transfer function to manipulate flow rate removes much of the transient fluctuations in the production rate. Furthermore, state variable feedback with pole placement improves the speed of response by about six times. 

Since it was desired not to lose the advantage already gained from using a first-order lag on Q, the scheme shown in Figure 6 was actually used for the pole placement tests. Figure 6 differs from Figure 5 only in that the Q lag of Figure 2 is included. 

Figure 7 shows some of the best results from these pole placement tests. The major difference between these curves and those of Figure 4 is the speed of response. The transient time has been reduced from 6 hours to about 1 hour. 

Pole placement. K is determined to assign the closed loop poles given by... 

Decoupling. K is determined to decouple some blocks of process/ manipulated variables. This is usually complemented by a decentralized control or a pole placement strategy. 

In any case, the general control structure is as depicted in Figure 14, where integral actions have been introduced to avoid the static errors. For pole placement, there are many options to implement the feedback law and additional requirements (such as decoupling, as already mentioned) can be settled. 

The purpose of this example is to show how to design the control system by using the pole-placement technique and the use of integrators. The integrators can be represented by introducing a new set of state variables u(t), so the equations of the global system are the following ... 

In accordance with the pole-placement technique, the linear time-invariant feedback control is defined as ... 

The centralized control can be approached using different techniques pole-placement, optimal control and loop decoupling. When the whole state is not accessible, a motivation to introduce a state observer is discussed. A detailed example when all state variables are accessible, i.e. when the state observer it is not necessary, has been explained. It is important to remark that the previously cited techniques are not widely used in CSTR control. This is due to the fact that these procedures require non-intuitive matrix tuning and computations, which are not familiar in the process industry. Nevertheless, for complex processes, these procedures can be the only solution to the control problem, when a limited set of sensors are available. 

Extended Luen-berger Observer Process model, (including process kinetics, but it is possible to estimate some kinetic parameters online), process inputs. Well known approach. It allows tuning the convergence rate by pole placement. Model hnearization Inputs knowledge Stability and convergence are only locally vahd. [6]... 

The primary interest in the pole placement literature recently has been in finding an analytical solution for the feedback matrix so that the closed loop system has a set of prescribed eigenvalues. In this context pole placement is often regarded as a simpler alternative than optimal control or frequency response methods. For a single control (r=l), the pole placement problem yields an analytical solution for full state feedback (e.g., (38), (39)). The more difficult case of output feedback pole placement for MIMO systems has not yet been fully solved(40). 

In the past few years, a number of workable pole-placement algorithms have been published. However, their application to MIMO systems with incomplete state variable feedback are often unsatisfactory in that ... 

For complete pole placement, it is usually required that r+n n+1, thus the total number of inputs and outputs are considerably larger than the minimum condition rxm n. Here the minimum condition means that when rxn n, it is likely that a solution exists for the resulting set of nonlinear equations. 

Note that a set of linear algebraic equations results. More details on pole placement with a dynamic controller have been reported by Brasch and Pearson (41). 

Davison, E., "On Pole Placement in Linear Systems with Incomplete State Feedback," IEEE Trans. Auto. Cont., 1970, AC-15, 348. 

Adaptive Pole placement controller is experimentally tested in a slot milling operation where the axial depth of cut is varied stepwise, as shown in Fig. 2. 

---