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

Abstract

SummaryLinear quadratic regulator (LQR), a popular technique for designing optimal state feedback controller, is used to derive a mapping between continuous and discrete time inverse optimal equivalence of proportional integral derivative (PID) control problem via dominant pole placement. The aim is to derive transformation of the LQR weighting matrix for fixed weighting factor, using the discrete algebraic Riccati equation (DARE) to design a discrete time optimal PID controller producing similar time response to its continuous time counterpart. Continuous time LQR-based PID controller can be transformed to discrete time by establishing a relation between the respective LQR weighting matrices that will produce similar closed loop response, independent of the chosen sampling time. Simulation examples of first/second order and first-order integrating processes exhibiting stable/unstable and marginally stable open loop dynamics are provided, using the transformation of LQR weights. Time responses for set-point and disturbance inputs are compared for different sampling times as fraction of the desired closed loop time constant.

Autonomous Collision Avoidance Using MPC with LQR-Based Weight Transformation

“…The objective of this strategy is to design the Q matrix from a discrete time algebraic Ricatti equation, which will ensure almost equal optimality between the DLQR and the unconstrained MPC control problem. It is noted that in this design the weighting factor R is chosen as a fixed value as in [ 19 ]. Moreover, the weighting matrix Q obtained from the inverse DLQR problem should be designed in such a way that the closed-loop poles are placed in the desired location within the unit circle by satisfying user-defined specifications using Ackerman’s pole placement formula.…”

“…In order to obtain the appropriate Q matrix, for designing an MPC-based reference tracking controller, we have chosen Q = diag(Q 11 , Q 22 , Q 33 ) for system (19) and the optimal control gainK with a fixed weighting factor R. For the reference tracking problem, MPC can be recast as a DLQT problem as described in Section 2.2. Therefore, it can be inferred that the weighting matrix Q obtained from the inverse DLQT problem will provide almost the same controller gain for MPC, which is described in the following Theorem.…”

“…Theorem 1. MPC will ensure the reference tracking for system (19), guaranteeing the suboptimality of the DLQT control problem by satisfying (2)-( 5) with Q T 1 = Q 1 ≥ 0, R T = R > 0 and P T 1 = P 1 > 0, where Q 1 = diag(Q, Q), R 1 = diag(R, R) and the controller gainŝ K = diag(K,K) with Q = diag(Q 11 , Q 22 , Q 33 ), and P 1 = diag(P, P) where P will be as (3)…”

“…Since the inputB d matrix in (19) includes only three non-zero elements, the three components of controller gains (k 2 , k 4 , k 6 ) will be zero, and the rest of the gains will be nonzero elements. The following section describes the system model required for designing the weighting matrix within the MPC controller.…”

“…However, these weighting matrices obtained from DLQR can also be used to compute the controller of unconstrained MPC [ 6 ]. Therefore, using the aforementioned concept, the objective of this paper is to design the weighting matrix Q of the DLQR in such a way that Q can be used within the MPC framework, while assuring asymptotical stability of the system [ 19 ]. DLQR and unconstrained MPC can be related through DARE, but using the weighting matrix obtained from the inverse optimal control problem of DLQR in MPC design cannot ensure solving the reference tracking problem.…”