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4F3 Predictive Control - Lecture 2 p 1/23 4F3 - Predictive Control Lecture 2 - Unconstrained Predictive Control Jan Maciejowski jmm@engcamacuk

4F3 Predictive Control - Lecture 2 p 2/23 References Predictive Control J M Maciejowski Predictive Control with Constraints, Prentice Hall UK, 2002 E F Camacho Model Predictive Control, Springer UK, 2 nd ed, 2003 Discrete Time Control G F Franklin et al Digital Control of Dynamic Systems, Adison Wesley 3 rd ed, 2003 K J Åstrom and B Wittenmark Computer Controlled Systems, Prentice Hall 3 rd ed, 2003

Standing Assumptions From now on, we will deal with the DT system: Assumptions: x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) z(k) = Hx(k) (A, B) is stabilizable and (C, A) detectable C = I state feedback H = C all outputs/states are controlled variables Goal is to regulate the states around the origin No delays, disturbances, model errors, noise etc Final two assumptions will be relaxed in final section of course 4F3 Predictive Control - Lecture 2 p 3/23

4F3 Predictive Control - Lecture 2 p 4/23 Linear Quadratic Regulator (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) k=0 The state weight Q 0 penalizes non-zero states The input weight R 0 penalizes non-zero inputs Usually Q and R are diagonal and positive definite

4F3 Predictive Control - Lecture 2 p 5/23 LQR Problems - Infinite Horizon The infinite horizon LQR problem has an infinite number of decision variables {u(0),u(1),,} A simple and closed form solution exists if Q is positive semidefinite (Q 0) R is positive definite (R 0) The pair (Q 1 2,A) is detectable (See Mini-tutorial 7 in JMM or the 4F2 course notes) We will solve a finite horizon version of the LQR problem with the same assumptions as above for use in RHC

4F3 Predictive Control - Lecture 2 p 6/23 Receding Horizon Control input output set point time k k + 1 time 1 Obtain measurement of current state x 2 Compute optimal finite horizon input sequence {u 0(x), u 1(x),,u N 1 (x)} 3 Implement first part of optimal input sequence κ(x) := u 0(x) 4 Return to step 1

4F3 Predictive Control - Lecture 2 p 7/23 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 V (x, (u 0,,u N 1 )) = 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 Important: V ( ) is a function of the initial state x and the first N inputs u i, and not the time index k or the predicted states x i

4F3 Predictive Control - Lecture 2 p 8/23 Finite Horizon Optimal Control Some terminology: The vector x i is the prediction of x(k + i) given the current state x(k) and the inputs u(k +i) = u i for all i = 0, 1,,N 1 The integer N is the control horizon The matrix P R n n is the terminal weight, with P 0 The stability and performance of a receding horizon control law based on this problem is determined by the parameters Q, R, P and N

4F3 Predictive Control - Lecture 2 p 9/23 Some Notation Define the stacked vectors U R Nm and X R Nn as: U := u 0 u 1 u 2, X := x 1 x 2 x 3, u N 1 x N Note that u i R m and x i R n, and that x = x 0 = x(k) is known Can define stacked outputs Y R Np and controlled variables Z R Nq in a similar way

4F3 Predictive Control - Lecture 2 p 10/23 More Notation The cost function is defined as N 1 ) V (x,u) := x T NPx N + (x Ti Qx i + u Ti Ru i The value function is defined as i=0 V (x) := min U V (x,u) The optimal input sequence is defined as U (x) := argmin V (x,u) U =: {u 0(x),u 1(x),,u N 1(x)}

4F3 Predictive Control - Lecture 2 p 11/23 Derivation of RHC Law Compute prediction matrices Φ and Γ such that X = Φx + ΓU Rewrite the cost function V ( ) in terms of x and U Compute the gradient U V (x,u) Set U V (x,u) = 0 and solve for U (x) The RHC control law is the first part of this optimal sequence: ( ) u 0(x) = I m 0 0 U (x) When there are no constraints, it is possible to do this analytically

4F3 Predictive Control - Lecture 2 p 12/23 Constructing the Prediction Matrices Want to find matrices Φ and Γ such that X = Φx + ΓU: x 1 = Ax 0 + Bu 0 x 2 = Ax 1 + Bu 1 x 3 = Ax 2 + Bu 2

4F3 Predictive Control - Lecture 2 p 12/23 Constructing the Prediction Matrices Want to find matrices Φ and Γ such that X = Φx + ΓU: x 1 = Ax 0 + Bu 0 x 2 = Ax 1 + Bu 1 x 3 = Ax 2 + Bu 2 x 1 = Ax 0 + Bu 0 x 2 = A(Ax 0 + Bu 0 ) + Bu 1 = A 2 x 0 + ABu 0 + Bu 1 x 3 = A 3 x 0 + A 2 Bu 0 + ABu 1 + Bu 2 x N = A N x 0 + A N 1 Bu 0 + + ABu N 2 + Bu N 1

4F3 Predictive Control - Lecture 2 p 13/23 Constructing the Prediction Matrices Collect terms to get: x 1 A x 2 := A 2 x 0 + x N A N B 0 0 AB B 0 A N 1 B A N 2 B B u 0 u 1 u N 1 Recalling that x := x 0, the prediction matrices Φ and Γ are: Φ := A A 2 A N B 0 0, Γ := AB B 0 A N 1 B A N 2 B B

Constructing the Cost Function Recall that the cost function is V (x, U):= x T NPx N + N 1 i=0 (x Ti Qx i + u Ti Ru i ) = x T 0 Qx 0 + x 1 x 2 x N T Q Q Q P x 1 x 2 x N + u 0 u 1 u N 1 T R R R R u 0 u 1 u N 1 Recalling x := x 0, these can be rewritten in matrix form as V (x,u) = x T Qx + X T ΩX + U T ΨU Note that: P 0 and Q 0 Ω 0 R 0 Ψ 0 4F3 Predictive Control - Lecture 2 p 14/23

4F3 Predictive Control - Lecture 2 p 15/23 Constructing the Cost Function Recall that: V (x,u) = x T Qx + X T ΩX + U T ΨU X = Φx + ΓU V (x,u) = x T Qx + (Φx + ΓU) T Ω(Φx + ΓU) + U T ΨU = x T Qx + x T Φ T ΩΦx + U T Γ T ΩΓU + U T ΨU + x T Φ T ΩΓU + U T Γ T ΩΦx = x T (Q + Φ T ΩΦ)x + U T (Ψ + Γ T ΩΓ)U + 2U T Γ T ΩΦx

4F3 Predictive Control - Lecture 2 p 16/23 Solving for U (x) Recall that: V (x,u) = 1 2 UT GU + U T Fx + x T (Q + Φ T ΩΦ)x where G := 2(Ψ + Γ T ΩΓ) 0, ( Ψ 0 and Ω 0) F := 2Γ T ΩΦ Important: This is a convex and quadratic function of U The unique and global minimum occurs at the point where V U (x,u) = GU + Fx = 0 The optimal input sequence is therefore U (x) = G 1 Fx

4F3 Predictive Control - Lecture 2 p 17/23 Receding Horizon Control Law The optimal input sequence is: U (x) = G 1 Fx The RHC law is defined by the first part of U (x): ( ) u 0(x) = I m 0 0 U (x) Define: K rhc = ( ) I m 0 0 G 1 F so that u = K rhc x This is a time invariant linear control law It approximates the optimal infinite horizon control law

4F3 Predictive Control - Lecture 2 p 18/23 Alternative Formulations Many variations on our predictive control formulation exist: Could optimize over predicted input changes u i = u i u i 1 V ( ) is then a function of x(k) and u k 1 The decision variables are { u 0, u 1,, u N 1 } Allows penalties on rapid control fluctuations Could allow a larger horizon for prediction than for control Inputs are constrained at the end of the control horizon, eg u i = Kx i or u i = 0 for all i N

4F3 Predictive Control - Lecture 2 p 19/23 Stability in Predictive Control Warning: The RHC law u(k) = K rhc x(k) might not be stabilizing Stability of the RHC law depends on the proper choice of the parameters Q, R, P and N It is not obvious how to do this Example: Consider the following open-loop unstable system with 2 states and 1 input: ( ) ( ) 1216 0055 002763 x(k + 1) = x(k) + 0221 09947 0002763 u(k)

4F3 Predictive Control - Lecture 2 p 20/23 Stability in Predictive Control Fix Q = P = I Compute ρ(a + BK rhc ) for various R and N: Vary N = 1,2,,50 Vary R = 0,002,,10 Plot: Black stable White unstable Not all values of R and N guarantee a stable closed loop we will return to this in Lecture 5

4F3 Predictive Control - Lecture 2 p 21/23 Equivalence between RHC and LQR The solution to the infinite horizon LQR problem is: where u(k) = K lqr x(k) K lqr = (B T PB + R) 1 B T PA and P is the solution to the Algebraic Riccati Equation (ARE): P = A T PA A T PB(B T PB + R) 1 B T PA + Q See 4F2 notes or JMM mini-tutorial 7 If the terminal weight P in the finite horizon cost function V ( ) is a solution to the ARE above, then: K rhc = K lqr

4F3 Predictive Control - Lecture 2 p 22/23 Implementation of the RHC Law K rhc u(k) DT System x(k) The matrix K rhc is a time-invariant, linear state feedback gain The RHC law is given by u(k) = K rhc x(k) Important: For our unconstrained problem, the control law can be written in closed-form

4F3 Predictive Control - Lecture 2 p 23/23 Output Feedback Predictive Control K rhc u(k) DT System y(k) ˆx(k k) Observer Controller If C I output feedback Use an observer to provide estimates ˆx(k k) RHC law is the same with x(k) replaced by ˆx(k k)