Prediktivno upravljanje primjenom matematičkog programiranja
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1 Prediktivno upravljanje primjenom matematičkog programiranja Doc. dr. sc. Mato Baotić Fakultet elektrotehnike i računarstva Sveučilište u Zagrebu
2 Outline Application Examples PredictiveControl of Linear Systems Receding Horizon Control Multi parametric Programming Optimal Control of Piecewise Affine Systems Multi parametric Integer Programming Dynamic Programming based algorithm Application Examples revisited Is that all? Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 2
3 Application Example 1: Adaptive Cruise Control Test Vehicle Traffic Scene Autonomous Driving IR Scanner Sensors Radar Virtual Traffic Scene Cameras Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 3
4 Sensor Fusion Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 4
5 Application Example 2 Electronic Throttle Main challenges Friction (gearbox) Limp-Home nonlinearity (return spring) Constraints Quantization (dual potentiometer + A/D) Sampling Time : 5 msec Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 5
6 Model Predictive Control (MPC) Determine/measure x(t) past future Use model to predict future behaviour of the system Compute optimal sequence of inputs over horizon Predicted outputs Manipulated u(t+k) Inputs t t+1 t+m t+p Implement first control input u(t) Wait for next sampling time; t:= t +1 t+1 t+2 t+1+m t+1+p Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 6
7 Model Predictive Control On Line Optimization obtain U* optimization problem plant state x apply u 0 * PLANT output y Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 7
8 Limitations of MPC On line application is limited to ʺslowʺ processes to ʺsimpleʺ process by available computing power Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 8
9 Optimal Control for Constrained Linear Systems System Discrete Linear Dynamics Constraints on the state Constraints on the input x(k) X u(k) U Objectives Optimal Performance Stability Feasibility (feedback is stabilizing) (feedback exists for all time) Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 9
10 Constrained Finite Time Optimal Control (CFTOC) of Linear Systems Linear Performance Index, p {1, } Constraints Algebraic manipulation Linear Program (LP) Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 10
11 Linear Program Standard form: Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 11
12 Constrained Finite Time Optimal Control (CFTOC) of Linear Systems Quadratic Performance Index Constraints Algebraic manipulation Quadratic Program (QP) Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 12
13 Quadratic Program Convex optimization if P º 0 Very hard problem if P ² 0 Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 13
14 Optimization Problem Formulation z: Decision Variables x: Parameters Definition: Feasible Set X f X f = { x R n z R s, g(z,x) 0} Value Function Optimizer J * (x), x X f z * (x), x X f Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 14
15 Receding Horizon Policy Off Line Optimization Optimization Problem u* ( x) Explicit Solution off-line (=Look-Up Table) u* = f(x) x control u* plant state x PLANT output y Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 15
16 Receding Horizon Policy Off Line Optimization Solve LP/QP for all x via multi-parametric Programming (mp-lp, mp QP) Piecewise affine state-feedback law k = T-1 k = 0 Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 16 k
17 Receding Horizon Policy Off Line Optimization Solve LP/QP for all x via multi-parametric Programming (mp-lp, mp QP) Piecewise affine state-feedback law Receding Horizon k = T-1 k = 0 Look-up table k Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 17
18 Multi parametric Programming z: Decision Variables x: Parameters Conventional math programming: Given x 0, solve optimization problem to obtain z * (x 0 ). Multi parametric Programming: Determine optimizer z * (x 0 ) for a range of parameters x 0 obtain explicit expression for z * (x). Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 18
19 Example: Solution of mp QP Karush Kuhn Tucker (KKT) conditions: Constraint i active G i z * W i S i x = 0, λ i* 0 Constraint j inactive G j z * W j S j x < 0, λ j* = 0 Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 19
20 Example: Solution of mp QP 1. Find local linear z * (x 0 ) 2. Find set where linear z * (x 0 ) is valid Critical Region 3. Proceed iteratively to find z * (x), x X f Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 20
21 Example: Solution of mp QP 1. Find local linear z * (x) KKT: From (1) : From (2) : solve QP for x 0 to find (z *,λ * ) identify active constraints G i z * W i S i x 0 = 0 (λ i* 0) form matrices by collecting active constraints In some neighborhood of x 0, λ and z are explicit linear functions of x Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 21
22 Example: Solution of mp QP 2. Find set where linear z * (x) is valid Critical Region Substitute and into the constraints: Polytopic Critical Region C 0 C 0 x space x 0 X Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 22
23 Example: Solution of mp QP 3. Proceed iteratively to find z * (x), x X f Pick up new points x i outside of C 0 x i C 0 Solve QP for new point x i Extract active constraints and compute C i x i Ci C 0 Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 23
24 Characteristics of mp LP Solution The optimizer z * (x) is continuous and piecewise affine (Note: if the optimizer z * (x) is not unique, there exists one with the above properties) The feasible set X f* is convex and partitioned into polyhedral regions The value function J * (x) is convex and piecewise affine Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 24
25 Characteristics of mp QP Solution The optimizer z * (x) is continuous and piecewise affine The feasible set X f* is convex and partitioned into polyhedral regions The value function J * (x) is convex, C 1 differentiable and piecewise quadratic Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 25
26 Outline Application Examples PredictiveControl of Linear Systems Receding Horizon Control Multi parametric Programming Optimal Control of Piecewise Affine Systems Multi parametric Integer Programming Dynamic Programming based algorithm Application Examples revisited Is that all? Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 26
27 Piecewise Affine (PWA) Systems if polyhedral partition of state+input space D state and input constraints are included in D D 1 D 3 D 2 D 4 PWA systems are equivalent to other hybrid system classes Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 27
28 CFTOC of PWA Systems 1/ Norm Parameters P, X f, T :influence unknown Stability: effect of parameters unclear k = 0 k = T-1 Solution to the CFTOC k where P ik, i=1,,r k is a polyhedral partition of the set X k of feasible states x(k) at time k with k =0,,T 1. Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 28
29 CFTOC of PWA Systems 1/ Norm Parameters P, X f, T :influence unknown Stability: effect of parameters unclear Receding Horizon k = 0 k = T-1 Solution to the CFTOC k where P ik, i=1,, R k, is a polyhedral partition of the set X k of feasible states x(k) at time k with k =0,,T 1. Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 29
30 CFTOC of PWA Systems 1/ Norm Introduce optimization vector with integer variables Multi parametric Mixed Integer Linear Program (mp MILP) Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 30
31 CFTOC of PWA Systems Efficient Algorithm Algorithm based on Dynamic programming recursion Multi parametric program solver (mp LP, mp QP) Basic polyhedral manipulation Comparison of PWA value functions over polyhedra Post processing Result Exact Globally optimal PWA feedback control law Look up table Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 31
32 PWA system Example PWA 2 PWA 1 1 Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 32
33 CFTOC Solution After 11 (or more) Steps Optimal Control u*(t) Value Function J* Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 33
34 PWA Systems Linear vs. Quadratic Cost The solution to to the optimal control problem is is a time varying PWA state feedback control law of of the form is is a partition of of the the set set of of feasible states x(k). x(k). k Qxk {1, } C i is polyhedral x Qx C i is bounded by quadratic functions Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 34
35 Outline Application Examples PredictiveControl of Linear Systems Receding Horizon Control Multi parametric Programming Optimal Control of Piecewise Affine Systems Multi parametric Integer Programming Dynamic Programming based algorithm Application Examples revisited Is that all? Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 35
36 Why Compute an Explicit Solution? 1. Understand the Controller Visualization Analysis x(0)in Region #12 Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 36
37 Why Compute an Explicit Solution? 2. Fast Implementation versus Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 37
38 Why Compute an Explicit Solution? 3. Cheap Implementation versus ~$10 (Look-Up-Table & μp) ~$10000 (PC & CPLEX) Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 38
39 Application Example 1: Adaptive Cruise Control Test Vehicle Traffic Scene Autonomous Driving IR Scanner Sensors Radar Virtual Traffic Scene Cameras Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 39
40 Application Example 1: Adaptive Cruise Control Objectives track reference speed respect minimum distance do not overtake on the right side of a neighboring car consider all objects on all lanes consider cost over future horizon Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 40
41 Experimental Results Vref=108km/h dmin=40m Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 41
42 Application Example 2 PWA Model of the Throttle State vector Friction: 5 affine parts Limp Home: 3 affine parts Zero Order Hold 15 (discrete time) PWA dynamics if Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 42
43 Application Example 2 Throttle Results Cost System Constraints Weights Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 43
44 Application Example 2 Throttle Results Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 44
45 Application Example 2 Throttle Results Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 45
46 Outline Application Examples PredictiveControl of Linear Systems Receding Horizon Control Multi parametric Programming Optimal Control of Piecewise Affine Systems Multi parametric Integer Programming Dynamic Programming based algorithm Application Example revisited Is that all? Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 46
47 There is much more Exploiting the solution structure Exploiting the problem structure Infinite Horizon (T ) Low complexity control Stability & feasibility analysis... Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 47
48 Conclusions All results and plots were obtained with the MPT toolbox MPT is a MATLAB toolbox that provides efficient code for (Non) Convex Polytope Manipulation Multi Parametric Programming Control of PWA and LTI systems Baotić :: Prediktivno upravljanje primjenom matematičkog programiranja 48
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