Pole placement based on derivative of states

Article history: Received 3 June 2016 Received in revised form 20 October 2016 Accepted 25 October 2016 State feedback is one of the important concepts in control theory. There are well defined methods like Ackermann’s formula and Bass-Gura formula to find required gain matrix (K) which place close loop poles at the desired location. Here, instead of states, derivative of states are used in order to find the suitable control law. One of the applications of this type of feedback is vibration suppression of mechanical systems where feedback signal is generally taken from an accelerometer. Effectiveness of the proposed method is shown by simulation.


Introduction
*When input is applied to a system, output comes out. Sometimes output is not as good as desired. Possible reasons may be slowness of response, stability problem or steady state errors (Chen, 2014;Ogata, 2010). In these cases, control is applied in order to make the response better. Fig. 1 shows a general schematic of a system: Here, there are m inputs and n outputs. H is a set of differential equations (ODE) which govern the output response. When system response is not desired, a control system must be added to the original system in order to make the overall response better. In order to do this, two control techniques can be used: It has been shown that feedback control has advantages over feedforward control (Ogata, 2010). Control problems can be divided into 2 groups (Chen, 2014;Ogata, 2010): 1. Tracking problems, 2. Regulation problems.
In tracking problems, goal is to follow a reference with minimal error. In regulation problems, system output must be keep at the desired level despite of disturbance and input changes.
In order to design a controller, either Laplace transform based methods or state space based methods can be used.
A lot of work has been done on pole placement problem by using state feedback (Valášek and Olgaç, 1999;Tuel, 1966). In this paper, pole placement problem is solved by derivative of state feedback.

State derivative based feedback control law
Assume a linear time invariant (LTI) system given by Eq. 1: and ( ) ∈ .
( ) is state vector and ( ) is control input. Assume pair (A, B) is controllable, i.e. controllability matrix is full rank (Eq. 2).
Eq. 2 uses derivative of states in order to produce control signal.

Method for determining gain matrix K
In order to solve the state feedback control problem for new system of Eq. 5, (pole placement) there are well known methods like (Chen, 2014): 1. Ackermann's Method, 2. Bass-Gura Method.

An example
In order to show effectiveness of proposed method, an example is given. Assume the circuit shown in Fig. 1. State space model is given by (Eq. 10): where, [ , 1 , 2 ] is state vector, is inductor current, 1 is capacitor 1 voltage and 2 is capacitor 2 voltage.
is assumed as control input (Fig. 2).

Simulation
In order to simulate the system Matlab ® / Simulink ® has used. Matlab ® has great variety of tools in order to simulate dynamical systems. Fig. 3 shows simulink diagram of the example studied before.
Results (state variable of circuit: , 1 , 2 ) are shown in Fig. 4. Fig. 4 shows that applying the control law given by = −10 . − 33 1 − 26 2 can force the system to return to equilibrium point [0, 0, 0] T , also close loop system has a more fast response than to original system. Although in simulation environments, all the variables are measurable easily, in a real life there are cases which only derivative of states can be obtained. For example in mechanical vibration suppression, usually an accelerometer is used as sensor and obtained states are velocity and acceleration not position and velocity. In order to solve these family of problems modification shown in these paper must be applied to formulas.

Conclusion
Pole placement problem is studied in recent decades by many researchers. In this paper, instead of states, derivative of states is used for placing poles at the desired location. A method is described for finding required gain matrix. Effectiveness of studied method is shown with an example.
Next step is to apply the studied method to an industrial application.