4F3 - Predictive Control

Transcription

1 4F3 Predictive Control - Lecture 3 p 1/21 4F3 - Predictive Control Lecture 3 - Predictive Control with Constraints Jan Maciejowski jmm@engcamacuk

2 4F3 Predictive Control - Lecture 3 p 2/21 Constraints on System Variables In practice, system variables are always constrained by: Physical limitations Input constraints eg actuator limits State constraints eg reservoir capacities Safety considerations (eg critical temperatures/pressures) Performance specifications (eg limit overshoot) constraints 1 y 1 4 output set point 1 y y y y 5 3 time

3 4F3 Predictive Control - Lecture 3 p 3/21 Systems with Input Saturation A common system nonlinearity is input saturation x(k + 1) = Ax(k) + B sat(u(k)) y(k) = Cx(k) (nonlinear) Easily transformed into a constraint on a linear system: u {i} u {i} ū {i} u sat (u) where v {i} is the i th component (row) of a column vector v

4 4F3 Predictive Control - Lecture 3 p 4/21 Constrained LQR Problem Problem: Given an initial state x(0) at time k = 0, compute and implement an input sequence {u(0),u(1),,} that minimizes the infinite horizon cost function ( ) x(k) T Qx(k) + u(k) T Ru(k) i=0 while guaranteeing that constraints are satisfied for all time It is usually impossible to solve this problem Predictive control provides an approximate solution RHC laws with constraints will be nonlinear

5 4F3 Predictive Control - Lecture 3 p 5/21 Constrained Finite Horizon Optimal Control Problem: Given an initial state x = x(k), compute a finite horizon input sequence {u 0,u 1,,u N 1 } that minimizes the finite horizon cost function x T NPx N + N 1 ) (x Ti Qx i + u Ti Ru i i=0 where x 0 = x x i+1 = Ax i + Bu i, i = 0, 1,,N 1 while guaranteeing that all constraints are satisfied over the prediction horizon i 0, 1,,N

6 4F3 Predictive Control - Lecture 3 p 6/21 Receding Horizon Control input output set point constraint constraint time k k + 1 time 1 Obtain measurement of current output/state 2 Compute optimal finite horizon input sequence subject to constraints 3 Implement first part of optimal input sequence 4 Return to step 1

7 4F3 Predictive Control - Lecture 3 p 6/21 Receding Horizon Control input output set point constraint constraint time k k + 1 time 1 Obtain measurement of current output/state 2 Compute optimal finite horizon input sequence subject to constraints 3 Implement first part of optimal input sequence 4 Return to step 1

8 4F3 Predictive Control - Lecture 3 p 6/21 Receding Horizon Control input output set point constraint constraint time k k + 1 time 1 Obtain measurement of current output/state 2 Compute optimal finite horizon input sequence subject to constraints 3 Implement first part of optimal input sequence 4 Return to step 1

9 4F3 Predictive Control - Lecture 3 p 6/21 Receding Horizon Control input output set point constraint constraint time k k + 1 time 1 Obtain measurement of current output/state 2 Compute optimal finite horizon input sequence subject to constraints 3 Implement first part of optimal input sequence 4 Return to step 1

10 4F3 Predictive Control - Lecture 3 p 7/21 Prediction Matrices Recall that we previously solved for the sequence of predicted states X in terms of the stacked inputs U: x 1 x 2 x N := A A 2 A N x 0 + B 0 0 AB B 0 A N 1 B A N 2 B B u 0 u 1 u N 1 or, defining x := x 0, X := Φx + ΓU The matrices Φ and Γ are the prediction matrices

11 4F3 Predictive Control - Lecture 3 p 8/21 Incorporating Constraints Now incorporate a set of linear inequality constraints on the predicted states x i and inputs u i M i x i + E i u i b i, for all i = 0, 1,,N 1 M N x N b N Many constraints take this form: M s = 0 Input constraints only E s = 0 State constraints only Can include constraints on outputs or controlled variables For simplicity, assume that E i = E, M i = M and b i = b for i = 0, 1,,N 1

12 4F3 Predictive Control - Lecture 3 p 9/21 Writing Constraints in Standard Form Suppose we have the following input and output constraints: u low u i u high, i = 0, 1,,N 1 y low y i y high, i = 0, 1,,N Recalling that y i = Cx i, this is equivalent to: 0 0 C +C x i + I +I 0 u i 0 u low +u high y low +y high for i = 0, 1,,N 1 Similar expression for terminal constraint (in terms of x N only)

13 4F3 Predictive Control - Lecture 3 p 10/21 Writing Constraints in Standard Form From the previous example, we can write the constraints in the form: by defining: M i x i + E i u i b i, for all i = 0, 1,,N 1 M N x N b N 0 I 0 +I M i :=, E i :=, b i := C 0 +C 0 u low +u high y low +y high, for i = 0, 1,,N 1 and M N := C, b N := +C y low +y high

14 Writing Constraints in Terms of x, X and U Taking all of the constraints together: M x M M N x 1 x N + E E N u 0 u N 1 b 0 b 1 u N By appropriately defining D, M, E and c (recalling x := x 0 ): Dx + MX + EU c Next, will eliminate X using prediction matrices 4F3 Predictive Control - Lecture 3 p 11/21

15 4F3 Predictive Control - Lecture 3 p 12/21 Writing Constraints in Terms of x and U Substitute X = Φx + ΓU into Dx + MX + EU c and collect terms The constraints can be written in the form: where and JU c + Wx J := MΓ + E W := D MΦ Our constraints are now in terms of the input sequence U and the initial state x = x 0 = x(k)

16 4F3 Predictive Control - Lecture 3 p 13/21 Writing Constraints in Terms of x and U In summary, the basic procedure is: Define linear inequalities in u i, x i, y i and z i Write the constraints in the form: M i x i + E i u i b i, for all i = 0, 1,,N 1 M N x N b N Stack the constraints to get them in the form: Dx + MX + EU c Substitute X = Φx + ΓU and rearrange to the form: JU c + Wx

17 4F3 Predictive Control - Lecture 3 p 14/21 Cost Function Recall that the cost function V (x,u) := x T NPx N + N 1 ) (x Ti Qx i + u Ti Ru i i=0 can be rewritten (with x := x 0 ) as V (x,u) = 1 2 UT GU + U T Fx + x T (Q + Φ T ΩΦ)x }{{} not dependent on U for some matrices F, G and Ω (defined in lecture 2) Remember that G 0 if P 0, Q 0 and R 0

18 4F3 Predictive Control - Lecture 3 p 15/21 Quadratic Programming Definition Quadratic Program (QP) Given matrices Q and A and vectors c and b, the optimization problem: subject to: is called a quadratic program (QP) 1 min θ 2 θt Qθ + c T θ Aθ b Proposition If Q 0 in the above quadratic program, then 1) The optimization problem is strictly convex 2) A global minimizer can always be found 3) The global minimizer is unique

19 4F3 Predictive Control - Lecture 3 p 16/21 Nature of Solutions to QPs θ 2 constrained minimum constraints Aθ b level set of cost function unconstrained minimum θ 1

20 4F3 Predictive Control - Lecture 3 p 17/21 Writing Problems as QPs Many problem types can be cast as QPs: Example: Nonlinear constraint subject to: 1 min θ 2 θt Qθ + c T θ max{e T θ,f T θ} b subject to: 1 min θ 2 θt Qθ + c T θ e T θ b, f T θ b Example: Nonlinear cost function subject to: min θ c T θ Aθ b subject to: min θ,δ δ Aθ b c T θ δ 0 c T θ δ 0

21 4F3 Predictive Control - Lecture 3 p 18/21 Constrained Optimal Control as a QP Our constrained optimal control problem is: min U subject to: 1 2 UT GU + U T Fx JU c + Wx This is a quadratic program in standard form Substitute θ U Q G c Fx A J b (c + Wx) The optimal solution is 1 A global minimum (when G 0) 2 Unique (when G 0)

22 4F3 Predictive Control - Lecture 3 p 19/21 Solution via Quadratic Programming Some QP parameters are functions of the current state x Without constraints, U (x) is a linear function of x With constraints, U (x) is nonlinear Must calculate U (x) by solving a QP online for each x See JMM 32 and 33 for an introduction to solving QPs In Matlab, our QP can be solved using U = quadprog(g,f*x,j,c+wx)

23 4F3 Predictive Control - Lecture 3 p 20/21 Implementing the RHC Law The RHC input is the first part of the optimal input sequence: ( ) κ rhc (x) := u 0(x) = I m 0 0 U (x) Since U (x) is no longer linear, κ rhc : R n R m is a nonlinear control law The dynamics of the closed loop system are nonlinear x(k + 1) = Ax(k) + Bκ rhc (x) κ rhc ( ) (QP Solver) u(k) DT System x(k)

24 4F3 Predictive Control - Lecture 3 p 21/21 Complexity of Solutions Example: Double integrator x(k + 1) = 1 1 x(k) + 0 u(k) Constraints: 6 4 Solution: Controller with 57 regions Each region i has u 0(x) = v i + K i x u 1, y 12 Horizon length = 12 Quadratic cost with Q = 1 0, R = 1, P = x {2} x {1}