Loop transfer recovery (LTR) procedures.ppt

Multivariable Control Systems,Ali KarimpourAssistant ProfessorFerdowsi University of Mashhad,,,2,Chapter 8,Multivariable Control System Design: LQG Method,Topics to be covered include:,LQG Control,Robustness Properties,Loop transfer recovery (LTR) procedures,- Recovering robustness at the plant output,- Recovering robustness at the plant input,Some practical consideration,- Shaping the principal gains (singular values),3,LQG Control,LQG Control,Robustness Properties,Loop transfer recovery (LTR) procedures,- Recovering robustness at the plant output,- Recovering robustness at the plant input,Some practical consideration,- Shaping the principal gains (singular values),4,LQG Control,In traditional LQG Control, it is assumed that the plant dynamics are linear and knownand that the measurement noise and disturbance signals (process noise) are stochastic with known statistical properties.,,,,That is, wd and wn are white noise processes with covariances,The problem is then to devise a feedback-control law which minimizes the ‘cost’,,,5,LQG Control,The solution to the LQG problem is prescribed by the separation theorem, which states that the optimal result is achieved by adopting the following procedure.,First, obtain an optimal estimate of the state x,,Then use this estimate as if it were an exact measurement of the state to solve the deterministic linear quadratic control problem.,,,6,LQG Control: Optimal state feedback,,The optimal solution for any initial state is,,where,,Where X=XT ≥ 0 is the unique positive-semidefinite solution of the algebraic Riccati equation,,7,LQG Control: Kalman filter,The Kalman filter has the structure of an ordinary state-estimator or observer, as,,,,Where Y=YT ≥ 0 is the unique positive-semidefinite solution of the algebraic Riccati equation,,8,LQG Control: Combined optimal state estimation and optimal state feedback,,,,,,,,Exercise 1: Proof the relation of KLQG(s) according to above figure.,9,Robustness Properties,LQG Control,Robustness Properties,Loop transfer recovery (LTR) procedures,- Recovering robustness at the plant output,- Recovering robustness at the plant input,Some practical consideration,- Shaping the principal gains (singular values),10,Robustness Properties,For an LQR-controlled system (i.e. assuming all the states are available and no stochastic inputs) it is well known (Kalman, 1994; Safonov and Athans, 1997) that, if the weight R is chosen to be diagonal, the sensitivity function,satisfies,,,,From this it can be shown that the system will have a gain margin equal to infinity, a gain reduction margin (lower gain margin) equal to 0.5 and a (minimum) phase margin of 60˚ in each plant input control channel.,Nyquist plot in MIMO case,11,Robustness Properties,,,12,Robustness Properties,Example 8-1: LQR design of a first order process.,,Consider a first order process,,,For a non-zero initial state the cost function to be minimized is,,The algebraic Riccati equation becomes,,,,,,,13,Robustness Properties,Example 8-1: LQR design of a first order process.,Consider a first order process,,,,,,,14,Robustness Properties,,,So, an LQR-controlled system has good stability margins at the plant inputs,,Arguments dual to those employed for the LQR-controlled system can then be used to show that, if the power spectral density matrix V is chosen to be diagonal, then at the input to the Kalman gain matrix Kf there will be an infinite gain margin, a gain reduction margin of 0.5 and a minimum-phase margin of 60˚.,15,Robustness Properties,,,So, an LQR-controlled system has good stability margins at the plant inputs,,And Kalman filter has good stability margins at the inputs to Kf,This was brought starkly to the attention of the control community by Doyle (1978 ) (in a paper entitled “Guaranteed Margins for LQR Regulators” with a very compact abstract which simply states “There are none”).,For an LQG-controlled system with a combined Kalman filter and LQR control law are there any guaranteed stability margins?,Unfortunately there are no guaranteed stability margins.,Doyle showed, by an example, that there exist LQG combinations with arbitrarily small gain margins.,16,Robustness Properties,Why there are no guaranteed stability margins in LQG controller.,,,,,,(Regulator transfer function),(Kalman Filter transfer function),guaranteed stability margins,guaranteed stability margins,The most important loop but no guaranteed stability margins,17,Loop Transfer Recovery,LQG Control,Robustness Properties,Loop transfer recovery (LTR) procedures,- Recovering robustness at the plant output,- Recovering robustness at the plant input,Some practical consideration,- Shaping the principal gains (singular values),18,Loop transfer recovery (LTR) procedures,,Assume that the plant model G(s) is minimum-phase and that it has at least as many inputs as outputs.,The LQG loop transfer function,,Guaranteed stability margins,If Kr in the LQR problem is designed to be large using the sensitivity recovery procedure of Kwakernaak (1969).,。