Lecture 14 - Model Predictive Control Part 1: The Concept

Transcription

1 Lecture 14 - Model Predictive Control Part 1: he Concept History and industrial application resource: Joe Qin, survey of industrial MPC algorithms Emerging applications State-based MPC Conceptual idea of MPC Optimal control synthesis Example Lateral control of a car Stability Lecture 15: Industrial MPC Control Engineering 14-1

2 MPC concept MPC = Model Predictive Control Also known as DMC = Dynamical Matrix Control GPC = Generalized Predictive Control RHC = Receding Horizon Control Control algorithms based on Numerically solving an optimization problem at each step Constrained optimization typically QP or LP Receding horizon control More details need to be worked out for implementation Control Engineering 14-2

3 Receding Horizon Control Receding Horizon Control concept future input trajectory RHC predicted future output prediction horizon Plant Plant Model current dynamic system states prediction horizon At each time step, compute control by solving an openloop optimization problem for the prediction horizon Apply the first value of the computed control sequence At the next time step, get the system state and re-compute Control Engineering 14-3

4 Current MPC Use Used in a majority of existing multivariable control applications echnology of choice for many new advanced multivariable control application Success rides on the computing power increase Has many important practical advantages Control Engineering 14-4

5 MPC Advantages Straightforward formulation, based on well understood concepts Explicitly handles constraints Explicit use of a model Well understood tuning parameters Prediction horizon Optimization problem setup Development time much shorter than for competing advanced control methods Easier to maintain: changing model or specs does not require complete redesign, sometimes can be done on the fly Control Engineering 14-5

6 History First practical application: DMC Dynamic Matrix Control, early 197s at Shell Oil Cutler later started Dynamic Matrix Control Corp. Many successful industrial applications heory (stability proofs etc) lagging behind 1-2 years. See an excellent resource on industrial MPC Joe Qin, Survey of industrial MPC algorithms history and formulations Control Engineering 14-6

7 Some Major Applications From Joe Qin, data, probably 1-2 order of magnitude growth by now Control Engineering 14-7

8 Emerging MPC applications Nonlinear MPC just need a computable model (simulation) NLP optimization Hybrid MPC discrete and parametric variables combination of dynamics and discrete mode change mixed-integer optimization (MILP, MIQP) Engine control Large scale operation control problems Operations management (control of supply chain) Campaign control Control Engineering 14-8

9 Emerging MPC applications Vehicle path planning and control nonlinear vehicle models world models receding horizon preview Control Engineering 14-9

10 Emerging MPC applications Spacecraft rendezvous with space station visibility cone constraint fuel optimality Underwater vehicle guidance From Richards & How, MI Missile guidance Control Engineering 14-1

11 State-based control synthesis Consider single input system for better clarity x ( t + 1) = Ax( + Bu( y ( = Cx( Infinite horizon optimal control τ = t+ 1 2 ( y( τ )) + r( u( τ ) u( τ 1) ) subject to : u( τ ) u Solution = Optimal Control Synthesis 2 min Control Engineering 14-11

12 State-based MPC concept x 2 Optimal control trajectories x 1 t > N Optimal control trajectories converge to (,) If N is large, the part of the problem for t > N can be neglected Infinite-horizon optimal control horizon-n optimal control Control Engineering 14-12

13 State-based MPC Receding horizon control; N-step optimal J x( t t N + = τ = t+ 1 2 ( y( τ )) + r( u( τ ) u( τ 1) ) subject to : u( τ ) u + 1) = Ax( + Bu( y( = Cx( Solution Optimal Control Synthesis, 2 min [ MPC Problem Solver] u( ) x( t Control Engineering 14-13

14 Control Engineering Predictive Model Predictive system model + + = ) ( 1) ( N t y t y Y M Model matrices + + = ) ( 1) ( N t u t u U M Fu HU Gx Y + + = initial condition response + control response x( x = Predicted output Future control input Current state (initial condition) = n CA CA G M = = (1) 1) ( ) ( (1) (2) (1) 3 2 h N h N h h h h B CA B CA CB H N N K M O M M K K K M O M M K K u( u = computed at the previous step + = 1) ( (3) (2) N h h h F M

15 Computations imeline Computed at the previous step u( x( u(t+1) x(t+1) Compute control based on x(. Apply it t t+1 time Assume that control u is applied and the state x is sampled at the same instant t Entire sampling interval is available for computing u Control Engineering 14-15

16 MPC Optimization Problem Setup MPC optimization problem J = Y Y + ru D subject to : U u Y = Gx + HU + Fu DU, min 1 st difference matrix 1 D = M 1 K 1 K M O K M his is a QP problem Solution [ MPC Problem, QP Solver] U u( t + 1) (1) x ( = U Control Engineering 14-16

17 QP solution QP Problem: AU b Q f = = J rd H = D U H QU H ( Gx + Fu) + f U min U = U( 1 I A = b = M u I 1 Predicted control sequence Standard QP codes can be used Control Engineering 14-17

18 Linear MPC Nonlinearity is caused by the constraints If constraints are inactive, the QP problem solution is U = Q his is linear state feedback u( t + 1) = l 1 f u = l ( ) 1 rd D + H H H ( Gx( + Fu( ) U l = 1 M u = z Kx + z 1 1 Su K 1 = l ( rd D + H H ) H G 1 S = l ( rd D + H H ) H F Can be analyzed as a linear system, e.g., check eigenvalues 1 z u = Kx 1 zx = Ax + Bu 1 Sz Control Engineering 14-18

19 Nonlinear MPC Stability heorem - from Bemporad et al (1994) Proof. Consider a MPC algorithm for a linear plan with constraints. Assume that there is a terminal constraint x(t + N) = for predicted state x and u(t + N) = for computed future control u If the optimization problem is feasible at time t, then the coordinate origin is stable. Use the performance index J as a Lyapunov function. It decreases along the finite feasible trajectory computed at time t. his trajectory is suboptimal for the MPC algorithm, hence J decreases even faster. Control Engineering 14-19

20 MPC Stability he analysis could be useful in practice heory says a terminal constraint is good MPC stability formulations (Mayne et al, Automatica, 2) erminal equality constraint erminal cost function Dual mode control infinite horizon erminal constraint set Increase feasibility region erminal cost and constraint set Control Engineering 14-2

21 Example: Lateral Control of a Car y lateral displacement u V Preview horizon Lane direction x Preview Control MacAdam s driver model (198) Consider predictive control design Simple kinematical model of a car driving at speed V x& = V cos a y& = V a& = u sin a lateral displacement steering Control Engineering 14-21

22 Lateral Control of a Car - Model y Lateral displacement y( a( u( V Assume a straight lane tracking a straight line Linearized system: assume a << 1 sin a a y& = Va cos a 1 a& = u Sampled-time equations (sampling time s ) a( t y( t + 1) = a( + + 1) = y( + u( s a( V s + u(.5v 2 s x Control Engineering 14-22

23 Lateral Control of a Car - MPC State-space system: x ( t + 1) = Ax( + Bu( a( 1 x( = A y( V Observation: ( Cx( Formulate predictive model Y = Gx + HU + Fu MPC optimization problem J =, 1 s B =.5V C = 1 = 2 s s y = [ ] ( Gx + HU + Fu) ( Gx + HU + Fu) subject to : Solution: U u, + ru D DU [ MPC QP] U u( t + 1) (1) x ( = U min Control Engineering 14-23

24 Impulse Responses 2 IMPULSE RESPONSE FOR LAERAL ERROR IMPULSE RESPONSE FOR HEADING ANGLE Control Engineering 14-24

25 Lateral Control of a Car - Simulation Simulation Results: HEADING ANGLE (deg) V = 5 mph Sample time of 2ms N = 2 All variables in SI units r = LAERAL ERROR SEERING CONROL (deg/s) IME (sec) Control Engineering 14-25

26 Control Design Issues Several important issues remain hey are not visible in this simulation Will be discussed in Lecture 15 (MPC, Part 2) All states might not be available Steady state error Need integrator feedback Large angle deviation linearized model deficiency introduce soft constraint Control Engineering 14-26