Robotics: Science and Systems

Transcription

1 Robotics: Science and Systems Model Predictive Control (MPC) Zhibin Li School of Informatics University of Edinburgh

2 Content Concepts of MPC MPC formulation Objective function and constraints Solving the QP problem Case study and comparison with LQR Summary of key MPC features 2

3 Model Predictive Control (MPC) 3

4 What is MPC Model predictive control (MPC) is: A modern control scheme that considers multiple control inputs and state/control constraints, eg actuation limits (power, torque, force), safety constraints (output joint motion/range) and so on. Optimization-based control that generates optimal trajectory as part of the feedback stabilization. A control approach that builds upon constrained optimization (QP) Also known as Receding Horizon Control RHC. 4

5 Concept of MPC Major steps: 1. Numerically solving an optimization problem at each control loop for the prediction horizon 2. Apply the first value of the computed control sequence 3. At the next control loop, obtain the system state feedback and repeat the above 2-step computation Principle of model predictive control. From: Dai, Li, et al. "Discrete-time model predictive control." Advances in Discrete Time Systems. InTech,

6 What are the differences? PID controllers do not have this predictive ability. Simple systems are often controlled well by generic PID controllers. LQR computes optimal gains, it can take multiple state variables into account. Both PID and LQR do not look ahead the future reference of the tasks during the computation of the control action now. However, MPC s control action is obtained by solving, at each loop, a finite horizon open-loop optimal control problem, using the current state as the initial state. So, the control action does not use pre-computed control gains. 6

7 Essential technique: QP Model predictive controllers compute optimal control action by: 1. Solving a quadratic program at each control loop; 2. QP solver converts a MPC optimization problem to a general form of quadratic programming problem. 7

8 MPC formulation Discrete State-Space (SS) model: D=0 in general case. We can transform it into a sequence of states X, for the next N time steps. Author of picture: Martin Behrendt 8

9 MPC formulation Transform SS into a sequence of states X and output Y, for the next N discrete time steps. Initial state x k at k that propagates through time. Sequence of control actions influence along time 9

10 MPC formulation 10

11 MPC formulation Y: sequence of future output (can be control target or constraints, depends on the output matrix C) U: sequence of control actions A t : state transition of initial state X 0 to Y (projection to each state) B t : resulted state transition of control input to Y 11

12 MPC control objective Intuitive understanding of objective: achieving best performance with minimum effort, ie optimality = best benefit cost ratio. Because the algorithm is optimizing a trajectory into the future, it can predict appropriate actions to avoid violating constraints in the future. Mathematically, objective function is where 12

13 Rewrite in QP formulation: objective The form of a QP problem: f(x)=1/2 x T Hx + c T x We need to formulate a control problem in the form of a QP problem. First, simplify the expression of error vector * This b is an intermediate variable here, it is not the same b in the constraint 13

14 Rewrite in QP formulation: objective Now this fits the form of a QP problem: f(x)=1/2 x T Hx + c T x 14

15 Rewrite in QP formulation: constraints Output constraints on Y: Reformulate the output in terms of control effort (principle: regulate control effort in a way that avoids hitting the output limit) 15

16 Quadratic Programming (QP) The objective of QP is to find vector x R n, that will minimize 1/2 x T Hx + c T x subject to A x b A eq x = b eq lb x ub This is a standard constrained Quadratic Programming (QP) problem! 16

17 Example of MPC in robot control Controlling a point mass system, the same as the previous LQR tracking example. 17

18 Previous example of LQR LQR tracking control with anti-windup. 18

19 Example of MPC and LQR What are the differences? 19

20 Summary Key features of MPC: Formulation has look-ahead time window to take future reference into account Problem formulation is typically structured in a QP form Open-loop optimization scheme Closed-loop is realized by repetitively using feedback of current state as the initial state for the optimization problem Principle of model predictive control. From: Dai, Li, et al. "Discrete-time model predictive control." Advances in Discrete Time Systems. InTech,

21 Summary Compared to MPC, LQR controller is an instantaneous controller, a hothead reasoner without looking at the future without considering the constraints while designing the optimal control law MPC, as a constrained optimizer, is more stable and optimal than simple controller that uses clipping or anti-windup technique to account for constraints. In cases of too limited control output or too large disturbance, the saturation is significant and lasts for long time, those simple techniques, clipping or anti-windup, may fail or becomes unstable. 21