Lecture 6: Deterministic Self-Tuning Regulators

Lecture 6: Deterministic Self-Tuning Regulators Feedback Control Design for Nominal Plant Model via Pole Placement Indirect Self-Tuning Regulators Direct Self-Tuning Regulators c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 1/19

Indirect Self-Tuning Regulators Consider a single input single output (SISO) system y(t) + a 1 y(t 1)+ + a n y(t n) = b 0 u(t d 0 ) + + b 1 u(t d 0 1) + + b m u(t d 0 m) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 2/19

Indirect Self-Tuning Regulators Consider a single input single output (SISO) system y(t) + a 1 y(t 1)+ + a n y(t n) = b 0 u(t d 0 ) + + b 1 u(t d 0 1) + + b m u(t d 0 m) Rewrite it in the form y(t) = a 1 y(t 1) a n y(t n) + b 0 u(t d 0 ) + + b 1 u(t d 0 1) + + b m u(t d 0 m) = φ(t 1) T θ c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 2/19

Indirect Self-Tuning Regulators Consider a single input single output (SISO) system y(t) + a 1 y(t 1)+ + a n y(t n) = b 0 u(t d 0 ) + + b 1 u(t d 0 1) + + b m u(t d 0 m) Rewrite it in the form y(t) = a 1 y(t 1) a n y(t n) + b 0 u(t d 0 ) + + b 1 u(t d 0 1) + + b m u(t d 0 m) = φ(t 1) T θ ] T where θ = [a 1,..., a n, b 0,..., b m [ φ(t 1) = y(t 1),..., y(t n), u(t d 0 ),..., u(t d 0 m) ] T c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 2/19

Recursive LS Algorithm Given the data { } N y(t), φ(t 1) 1 defined by the model y(t) = φ(t 1) T θ 0, t = 1, 2,..., N c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 3/19

Recursive LS Algorithm Given the data { } N y(t), φ(t 1) 1 defined by the model y(t) = φ(t 1) T θ 0, t = 1, 2,..., N Given ˆθ(t 0 ) and P(t 0 ) = (Φ(t 0 ) T Φ(t 0 )) 1, the LS estimate satisfies the recursive equations ˆθ(t) = ˆθ(t 1) + K(t) ( ) y(t) φ(t) T ˆθ(t 1) K(t) = P(t)φ(t) = P(t 1)φ(t) ( ) 1 1 + φ T P(t 1)φ(t) P(t) = P(t 1) P(t 1)φ(t) ( 1 + φ T P(t 1)φ(t)) 1 φt P(t 1) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 3/19

Algorithm using RLS and MD Pole Placement Off-line Parameters: Given polynomials B m (q), A m (q), A o (q) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 4/19

Algorithm using RLS and MD Pole Placement Off-line Parameters: Given polynomials B m (q), A m (q), A o (q) Step 1: Estimate the coefficients of A(q) and B(q), i.e. θ, using the Recursive Least Squares algorithm. Step 2: Apply the Minimum Degree Pole Placement algorithm to compute R(q), T(q), S(q) with A(q), B(q) taken from the previous Step. c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 4/19

Algorithm using RLS and MD Pole Placement Off-line Parameters: Given polynomials B m (q), A m (q), A o (q) Step 1: Estimate the coefficients of A(q) and B(q), i.e. θ, using the Recursive Least Squares algorithm. Step 2: Apply the Minimum Degree Pole Placement algorithm to compute R(q), T(q), S(q) with A(q), B(q) taken from the previous Step. Step 3: Compute the control variable by R(q)u(t) = T(q)u c (t) S(q)y(t) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 4/19

Algorithm using RLS and MD Pole Placement Off-line Parameters: Given polynomials B m (q), A m (q), A o (q) Step 1: Estimate the coefficients of A(q) and B(q), i.e. θ, using the Recursive Least Squares algorithm. Step 2: Apply the Minimum Degree Pole Placement algorithm to compute R(q), T(q), S(q) with A(q), B(q) taken from the previous Step. Step 3: Compute the control variable by R(q)u(t) = T(q)u c (t) S(q)y(t) Repeat Steps 1, 2, 3 c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 4/19

Example 3.4 (Recursive Modification of Example 3.1) Given a continuous time system ÿ + ẏ = u, c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 5/19

Example 3.4 (Recursive Modification of Example 3.1) Given a continuous time system ÿ + ẏ = u, its pulse transfer function with sampling period h = 0.5 sec is G(q) = b 1q + b 2 q 2 + a 1 q + a 2 = 0.1065q + 0.0902 q 2 1.6065q + 0.6065 The task is to synthesize a 2-degree-of-freedom controller such that the complementary sensitivity transfer function is B m (q) A m (q) = 0.1761q q 2 1.3205q 0.4966 c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 5/19

Example 3.4 (Recursive Modification of Example 3.1) We have found the controller via MD pole placement R(q)u(t) = T(q)u c (t) S(q)y(t) with R(q) = q + 0.8467 S(q) = 2.6852 q 1.0321 T(q) = 1.6531 q that stabilizes both the discrete and continuous-time plants c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 6/19

Example 3.4 (Recursive Modification of Example 3.1) We have found the controller via MD pole placement with R(q)u(t) = T(q)u c (t) S(q)y(t) R(q) = q + 0.8467 S(q) = 2.6852 q 1.0321 T(q) = 1.6531 q that stabilizes both the discrete and continuous-time plants Let us design the adaptive controller which would recover the performance for the case when parameters of the plant - the polynomials A(q) and B(q) are not known! c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 6/19

2 1.5 1 0.5 output 0 0.5 1 1.5 2 0 10 20 30 40 50 60 70 80 90 time The response of the closed loop system with adaptive controller c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 7/19

5 4 3 2 1 control 0 1 2 3 4 5 0 10 20 30 40 50 60 70 80 90 time The control signal c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 8/19

1 a 1 0 1 2 0 5 10 15 20 25 30 35 40 1 a 2 0.5 0 0.5 0 5 10 15 20 25 30 35 40 0.25 b 1 0.2 0.15 0.1 0 5 10 15 20 25 30 35 40 0.4 b 2 0.2 0 0 5 10 15 20 25 30 35 40 time (sec) Adaptation of parameters of polynomials A(q) and B(q) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 9/19

Direct Self-Tuning Regulators The previous algorithm consists of two steps Computing estimates for the plant polynomials A(q), B(q) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 11/19

Direct Self-Tuning Regulators The previous algorithm consists of Step 1: Computing estimates Â(q), ˆB(q) for A(q), B(q) Step 2: Computing a controller R(q), T(q), S(q) based on Â(q), ˆB(q). Such procedure has potential drawbacks like Redundant step in design, why we should know A(q) and B(q)? Control design methods (pole placement, LQR,... ) are sensitive to mistakes in Â(q), ˆB(q). How to control such poor conditioned numerical computations? c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 11/19

Direct Self-Tuning Regulators The previous algorithm consists of Step 1: Computing estimates Â(q), ˆB(q) for A(q), B(q) Step 2: Computing a controller R(q), T(q), S(q) based on Â(q), ˆB(q). Such procedure has potential drawbacks like Redundant step in design, why we should know A(q) and B(q)? Control design methods (pole placement, LQR,... ) are sensitive to mistakes in Â(q), ˆB(q). How to control such poor conditioned numerical computations? I.e. Â(q) = q(q 1), ˆB(q) = q + ε c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 11/19

Direct Self-Tuning Regulators The idea of new approach comes from the observation that for the plant A(q)y(t) = B(q)u(t) = B + (q)b (q)u(t) and for the target system A m (q)y(t) = B m (q)u c (t) = B mp (q)b (q)u c (t) the Diophantine equation A o (q)a m (q) = A(q)R(q) + B (q)s(q) can be seen as a regression model to estimate coefficients R, S c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 12/19

Direct Self-Tuning Regulators The idea of new approach comes from the observation that for the plant A(q)y(t) = B(q)u(t) = B + (q)b (q)u(t) and for the target system A m (q)y(t) = B m (q)u c (t) = B mp (q)b (q)u c (t) the Diophantine equation A o (q)a m (q) = A(q)R(q) + B (q)s(q) can be seen as a regression model to estimate coefficients R, S A o (q)a m (q)y(t) = [ ] A(q)R p (q) + B (q)s(q) y(t) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 12/19

Direct Self-Tuning Regulators The idea of new approach comes from the observation that for the plant A(q)y(t) = B(q)u(t) = B + (q)b (q)u(t) and for the target system A m (q)y(t) = B m (q)u c (t) = B mp (q)b (q)u c (t) the Diophantine equation A o (q)a m (q) = A(q)R(q) + B (q)s(q) can be seen as a regression model to estimate coefficients R, S A o (q)a m (q)y(t) = A(q)R p (q)y(t) + B (q)s(q)y(t) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 12/19

Direct Self-Tuning Regulators The idea of new approach comes from the observation that for the plant A(q)y(t) = B(q)u(t) = B + (q)b (q)u(t) and for the target system A m (q)y(t) = B m (q)u c (t) = B mp (q)b (q)u c (t) the Diophantine equation A o (q)a m (q) = A(q)R(q) + B (q)s(q) can be seen as a regression model to estimate coefficients R, S A o (q)a m (q)y(t) = R p (q)a(q)y(t) + B (q)s(q)y(t) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 12/19

Direct Self-Tuning Regulators The idea of new approach comes from the observation that for the plant A(q)y(t) = B(q)u(t) = B + (q)b (q)u(t) and for the target system A m (q)y(t) = B m (q)u c (t) = B mp (q)b (q)u c (t) the Diophantine equation A o (q)a m (q) = A(q)R(q) + B (q)s(q) can be seen as a regression model to estimate coefficients R, S A o (q)a m (q)y(t) = R p (q)b(q)u(t) + B (q)s(q)y(t) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 12/19

Direct Self-Tuning Regulators The idea of new approach comes from the observation that for the plant A(q)y(t) = B(q)u(t) = B + (q)b (q)u(t) and for the target system A m (q)y(t) = B m (q)u c (t) = B mp (q)b (q)u c (t) the Diophantine equation A o (q)a m (q) = A(q)R(q) + B (q)s(q) can be seen as a regression model to estimate coefficients R, S A o (q)a m (q)y(t) = R p (q) B + (q) B (q)u(t)+b (q)s(q)y(t) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 12/19

Direct Self-Tuning Regulators The idea of new approach comes from the observation that for the plant A(q)y(t) = B(q)u(t) = B + (q)b (q)u(t) and for the target system A m (q)y(t) = B m (q)u c (t) = B mp (q)b (q)u c (t) the Diophantine equation A o (q)a m (q) = A(q)R(q) + B (q)s(q) can be seen as a regression model to estimate coefficients R, S A o (q)a m (q)y(t) = B (q) ( ) R(q) u(t) + S(q)y(t) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 12/19

Direct Self-Tuning Regulators for Minimum-Phase Systems Suppose that B(q) is stable and can be canceled, i.e. A(q)y(t) = B(q)u(t) = b 0 u(t d 0 ) + b 1 u(t d 0 1) + + b m u(t d 0 m) = B + (q)b (q)u(t) = B + (q)b 0 u(t) then Diophantine equation A o (q)a m (q) = A(q)R(q) + B (q)s(q) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 13/19

Direct Self-Tuning Regulators for Minimum-Phase Systems Suppose that B(q) is stable and can be canceled, i.e. A(q)y(t) = B(q)u(t) = b 0 u(t d 0 ) + b 1 u(t d 0 1) + + b m u(t d 0 m) = B + (q)b (q)u(t) = B + (q)b 0 u(t) then Diophantine equation A o (q)a m (q) = A(q)R(q) + B (q)s(q) leads to the regression ) A o (q)a m (q)y(t) = b 0 (R(q)u(t) + S(q)y(t) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 13/19

Direct Self-Tuning Regulators for Minimum-Phase Systems Suppose that B(q) is stable and can be canceled, i.e. A(q)y(t) = B(q)u(t) = b 0 u(t d 0 ) + b 1 u(t d 0 1) + + b m u(t d 0 m) = B + (q)b (q)u(t) = B + (q)b 0 u(t) then Diophantine equation A o (q)a m (q) = A(q)R(q) + B (q)s(q) leads to the regression ) A o (q)a m (q)y(t) = b 0 (R(q)u(t) + S(q)y(t) which is ) η(t) = b 0 (R(q)u(t) + S(q)y(t) = φ(t 1) T θ c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 13/19

Direct Self-Tuning Regulators for Minimum-Phase Systems In the formula η(t) = b 0 R(q)u(t) + b 0 S(q)y(t) = φ(t) T θ the vector of regressors is φ(t) = [u(t),..., u(t l), y(t),..., y(t l)] T and the vector of parameters is θ = [b 0 r 0,..., b 0 r l, b 0 s 0,..., b 0 s l ] T c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 14/19

Direct Self-Tuning Regulators for Minimum-Phase Systems In the formula η(t) = b 0 R(q)u(t) + b 0 S(q)y(t) = φ(t) T θ the vector of regressors is φ(t) = [u(t),..., u(t l), y(t),..., y(t l)] T and the vector of parameters is θ = [b 0 r 0,..., b 0 r l, b 0 s 0,..., b 0 s l ] T When θ is estimated by ˆθ, then R(q) = q l + ˆθ 2 ˆθ 1 q l 1 +..., S(q) = q l + ˆθ l+2 ˆθ 1 q l 1 +... c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 14/19

Direct Self-Tuning Regulators for Minimum-Phase Systems Some modification to the regression model A o (q)a m (q)y(t) = η(t) = b 0 R(q)u(t)+b 0 S(q)y(t) = φ(t) T θ can be made if we introduce the signals u f (t) = 1 A o (q)a m (q) u(t), y f(t) = 1 A o (q)a m (q) y(t) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 15/19

Direct Self-Tuning Regulators for Minimum-Phase Systems Some modification to the regression model A o (q)a m (q)y(t) = η(t) = b 0 R(q)u(t)+b 0 S(q)y(t) = φ(t) T θ can be made if we introduce the signals u f (t) = 1 A o (q)a m (q) u(t), y f(t) = 1 A o (q)a m (q) y(t) Then the regression model is y(t) = b 0 R(q)u f (t) + b 0 S(q)y f (t) = R(q)u f (t) + S(q)y f (t) = φ(t) T θ c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 15/19

Direct Self-Tuning Regulators: Summary Given polynomials A m (q), B m (q), A o (q) the relative degree d 0 of the plant c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 16/19

Direct Self-Tuning Regulators: Summary Given polynomials A m (q), B m (q), A o (q) the relative degree d 0 of the plant Step 1: Estimate coefficients of polynomials R(q) and S(q) from the regression model y(t) = b 0 R(q)u f (t) + b 0 S(q)y f (t) = φ(t) T θ c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 16/19

Direct Self-Tuning Regulators: Summary Given polynomials A m (q), B m (q), A o (q) the relative degree d 0 of the plant Step 1: Estimate coefficients of polynomials R(q) and S(q) from the regression model y(t) = b 0 R(q)u f (t) + b 0 S(q)y f (t) = φ(t) T θ Step 2: Compute the control signal by R(q)u(t) = T(q)u c (t) S(q)y(t) where T(q) = A o (q)a m (1) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 16/19

Direct Self-Tuning Regulators: Summary Given polynomials A m (q), B m (q), A o (q) the relative degree d 0 of the plant Step 1: Estimate coefficients of polynomials R(q) and S(q) from the regression model y(t) = b 0 R(q)u f (t) + b 0 S(q)y f (t) = φ(t) T θ Step 2: Compute the control signal by R(q)u(t) = T(q)u c (t) S(q)y(t) where T(q) = A o (q)a m (1) Repeat Steps 1 and 2 c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 16/19

Direct ST Regulators for Non-Minimum-Phase Systems If B(q) is not stable or cannot be canceled, i.e. A(q)y(t) = B(q)u(t) = b 0 u(t d 0 ) + b 1 u(t d 0 1) + + b m u(t d 0 m) = B + (q)b (q)u(t), B (q) Const then Diophantine equation A o (q)a m (q) = A(q)R(q) + B (q)s(q) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 17/19

Direct ST Regulators for Non-Minimum-Phase Systems If B(q) is not stable or cannot be canceled, i.e. A(q)y(t) = B(q)u(t) = b 0 u(t d 0 ) + b 1 u(t d 0 1) + + b m u(t d 0 m) = B + (q)b (q)u(t), B (q) Const then Diophantine equation A o (q)a m (q) = A(q)R(q) + B (q)s(q) leads to the regression A o (q)a m (q)y(t) = B (q) ( ) R(q)u(t) + S(q)y(t) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 17/19

Direct ST Regulators for Non-Minimum-Phase Systems If B(q) is not stable or cannot be canceled, i.e. A(q)y(t) = B(q)u(t) = b 0 u(t d 0 ) + b 1 u(t d 0 1) + + b m u(t d 0 m) = B + (q)b (q)u(t), B (q) Const then Diophantine equation A o (q)a m (q) = A(q)R(q) + B (q)s(q) leads to the regression A o (q)a m (q)y(t) = B (q) ( ) R(q)u(t) + S(q)y(t) which is nonlinear in parameters and has unstable factor η(t) = B (q) ( ) R(q)u(t) + S(q)y(t) = φ(t 1) T θ c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 17/19

Direct Self-Tuning Regulators: Summary Given polynomials A m (q), B m (q), A o (q) the relative degree d 0 of the plant c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 18/19

Direct Self-Tuning Regulators: Summary Given polynomials A m (q), B m (q), A o (q) the relative degree d 0 of the plant Step 1: Estimate coefficients of R(q) and S(q) from y(t) = R(q)u f (t) + S(q)y f (t) = φ(t) T θ Cancel possible common factors of R(q) and S(q) to compute R(q) and S(q) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 18/19

Direct Self-Tuning Regulators: Summary Given polynomials A m (q), B m (q), A o (q) the relative degree d 0 of the plant Step 1: Estimate coefficients of R(q) and S(q) from y(t) = R(q)u f (t) + S(q)y f (t) = φ(t) T θ Cancel possible common factors of R(q) and S(q) to compute R(q) and S(q) Step 2: Compute the control signal by R(q)u(t) = T(q)u c (t) S(q)y(t) where T(q) = A o (q)b mp (q) c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 18/19

Direct Self-Tuning Regulators: Summary Given polynomials A m (q), B m (q), A o (q) the relative degree d 0 of the plant Step 1: Estimate coefficients of R(q) and S(q) from y(t) = R(q)u f (t) + S(q)y f (t) = φ(t) T θ Cancel possible common factors of R(q) and S(q) to compute R(q) and S(q) Step 2: Compute the control signal by R(q)u(t) = T(q)u c (t) S(q)y(t) where T(q) = A o (q)b mp (q) Repeat Steps 1 and 2 c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 18/19

Tasks: Reading Chapter 3 of the book c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 19/19

Tasks: Reading Chapter 3 of the book Solve problems 3.1, 3.2, 3.3, 3.4 c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 19/19

Tasks: Reading Chapter 3 of the book Solve problems 3.1, 3.2, 3.3, 3.4 Implement Matlab-code for Examples 3.4 and 3.5 c Anton Shiriaev. May, 2007. Lectures on Adaptive Control: Lecture 6 p. 19/19