Pole Placement for Single-Input Linear System by Proportional-Derivative State Feedback

Abstract

This paper deals with the direct solution of the pole placement problem for single-input linear systems using proportional-derivative (PD) state feedback. This problem is always solvable for any controllable system. The explicit parametric expressions for the feedback gain controllers are derived which describe the available degrees of freedom offered by PD state feedback. These freedoms are utilized to obtain closed-loop systems with small gains. Its derivation is based on the transformation of linear system into control canonical form by a special coordinate transformation. The solving procedure results into a formula similar to Ackermann's one. In the present work, both time-invariant and timevarying linear systems are treated. The effectiveness of the proposed method is demonstrated by the simulation examples of both time-invariant and time-varying systems.

Transformation of LQR Weights for Discretization Invariant Performance of PI/PID Dominant Pole Placement Controllers

“…In equation (25), the Riccati solution and weights   P,Q, R are in the same form as of the continuous time cases in equation 12, (13). Similar to the earlier continuous time treatment, the discrete time optimal state feedback controller gain matrix s K can be achieved as (26) using the matrix P that minimizes the discrete quadratic cost function (24).…”

“…However, in order to obtain the same discrete PI/PID controller gains from DARE (25) as in continuous time version, here we derive the analytical expressions of the discrete time transformed LQR weighting matrix and discrete Riccati matrix solution   , QP as shown in (30)-(34), (48)-(51) and (68)-(70) for the generalized second order (1), first order integrating (35) and first order (52) system templates respectively, while keeping R fixed and with arbitrary sampling time Ts, which are the main contributions of this paper. Then DARE (25) is also solved using the matrices   , QP as in (30)-(34), (48)-(51) and (68)-(70) for second order, first order integrating and first order test-bench processes respectively as reported in Table 1, with the sampling time taken as Ts = 0.05 sec and discrete/continuous time LQR weighting factor 1  RR which produce the same PI/PID controller gains as with its continuous time version and shown in Table 2. Thus, it can be observed that the choice of sampling time in discrete LQR has minimal effect to obtain the same PI/PID controller gains as in the continuous time version up to Ts/τcl = 0.5 and consequently Ts = 0.05 sec for desired τcl = 0.1 (i.e.…”

“…The diagonal elements Q 11 , Q 22 of the weighting matrix Q using Eqs. (25), (60), and (65)-(68) are obtained as:…”

“…The optimum weight selection has also been extended for the noisy tracking (integral) problem using linear quadratic Gaussian (LQG) control with loop transfer recovery (LTR) for improved stability margins [24]. Also, optimal statefeedback PD controller design has been previously studied with minimum norm controller gain to reduce noise amplification in [25], [26], [27]. But previous literatures shown above, do not provide the transformation to directly map time domain specifications (usually adopted in dominant pole placement design) onto the discrete time LQR weights, which is the main contribution of this paper over existing literatures.…”

Stabilization of linear time-varying systems using proportional-derivative state feedback

“…Therefore, some researchers have proposed some approaches to study the performance analysis and design problems of PD control systems. In more recent work, a new technique for designing PD state controllers based on the concept of pole assignment control for single-input linear system is developed (Abdelaziz, 2015a, 2015b).…”