Linear Optimal Control. How does this guy remain upright?
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1 Linear Optimal Control How does this guy remain upright?
2 Overview 1. expressing a linear system in state space form 2. discrete time linear optimal control (LQR) 3. linearizing around an operating point 4. linear model predictive control 5. LQR variants 6. model predictive control for non-linear systems
3 A simple system k b m Force exerted by the spring: Force exerted by the damper: Force exerted by the inertia of the mass:
4 A simple system k m b Consider the motion of the mass there are no other forces acting on the mass therefore, the equation of motion is the sum of the forces: This is called a linear system. Why?
5 A simple system k Let's express this in ''state space form'': m b
6 A simple system k Let's express this in ''state space form'': m b
7 A simple system k Let's express this in ''state space form'': m b
8 A simple system k Let's express this in ''state space form'': m b
9 A simple system k Let's express this in ''state space form'': m b where
10 A simple system k m f Your finger b Suppose that you apply a force:
11 Suppose that you apply a force: A simple system
12 A simple system Suppose that you apply a force: Canonical form for a linear system
13 Continuous time vs discrete time Continuous time Discrete time
14 Continuous time vs discrete time Continuous time Discrete time What are A and B now?
15 Continuous time vs discrete time Continuous time Discrete time What are A and B now?
16 Simple system in discrete time We want something in this form:
17 Simple system in discrete time We want something in this form:
18 Simple system in discrete time We want something in this form:
19 Simple system in discrete time We want something in this form:
20 Continuous time vs discrete time CT DT CT DT
21 Exercise: write DT system dynamics External force Viscous damping
22 Exercise: write DT system dynamics Something else...
23 Overview 1. expressing a linear system in state space form 2. discrete time linear optimal control (LQR) 3. linearizing around an operating point 4. linear model predictive control 5. LQR variants 6. model predictive control for non-linear systems
24 The linear control problem Given: System:
25 The linear control problem Given: System: Cost function: where:
26 The linear control problem Given: System: Cost function: where:
27 The linear control problem Given: System: Cost function: where: Initial state: Calculate: U that minimizes J(X,U)
28 The linear control problem Given: System: Cost function: Important problem! How where: do we solve it? Initial state: Calculate: U that minimizes J(X,U)
29 One solution: least squares
30 One solution: least squares
31 One solution: least squares where
32 One solution: least squares where:
33 One solution: least squares Given: System: Cost function: where: Initial state: Calculate: U that minimizes J(X,U)
34 One solution: least squares Given: System: Cost function: Initial state: Calculate: U that minimizes J(X,U)
35 One solution: least squares Substitute X into J: Minimize by setting dj/du=0: Solve for U:
36 What can this do? Start here End here at time=t Solve for optimal trajectory: Image: van den Berg, 2015
37 What can this do? This is cool, but... only works for finite horizon problems doesn't account for noise requires you to invert a big matrix
38 Let's try to solve this another way Bellman optimality principle: Why is this equation true?
39 Let's try to solve this another way Bellman optimality principle: Cost-to-go from state x at time t Cost-to-go from state (Ax+Bu) at time t+1 Cost incurred on this time step Cost incurred after this time step
40 Let's try to solve this another way For the sake of argument, suppose that the cost-to-go is always a quadratic function like this: where:
41 Let's try to solve this another way For the sake of argument, suppose that the cost-to-go is always a quadratic function like this: where: Then:
42 Let's try to solve this another way For the sake of argument, suppose that the cost-to-go is always a quadratic function like this: where: Then: How do we minimize this term? take derivative and set it to zero.
43 Let's try to solve this another way How do we minimize this term? take derivative and set it to zero. optimal control as a function of state but: it depends on P_{t+1}...
44 Let's try to solve this another way How do we minimize this term? take derivative and set it to zero. How solve for P_{t+1}??? optimal control as a function of state but: it depends on P_{t+1}...
45 Let's try to solve this another way Substitute u into V_t(x):
46 Let's try to solve this another way Substitute u into V_t(x):
47 Let's try to solve this another way Substitute u into V_t(x):
48 Let's try to solve this another way Substitute u into V_t(x):
49 Let's try to solve this another way Substitute u into V_t(x): Dynamic Riccati Equation
50 Example: planar double integrator Build the LQR controller for: Initial position of the puck Goal position Initial velocity m=1 b=0.1 u=applied force Initial state: Time horizon: Cost fn: Air hockey table
51 Example: planar double integrator Step 1: Calculate P backward from T: P_100, P_99, P_98,, P_1 HOW? Air hockey table
52 Example: planar double integrator Step 1: Calculate P backward from T: P_100, P_99, P_98,, P_1 Air hockey table
53 Example: planar double integrator Step 1: Calculate P backward from T: P_100, P_99, P_98,, P_1 Air hockey table
54 Example: planar double integrator Step 1: Calculate P backward from T: P_100, P_99, P_98,, P_1 Air hockey table
55 Example: planar double integrator Step 1: Calculate P backward from T: P_100, P_99, P_98,, P_1 Air hockey table......
56 Example: planar double integrator Step 2: Calculate u starting at t=1 and going forward to t=t-1 Air hockey table
57 Example: planar double integrator origin
58 Example: planar double integrator u_x, u_y t
59 Example: planar double integrator
60 Example: planar double integrator origin 0
61 Example: planar double integrator origin 0
62 The infinite horizon case So far: we have optimized cost over a fixed horizon, T. optimal if you only have T time steps to do the job But, what if time doesn't end in T steps? One idea: at each time step, assume that you always have T more time steps to go this is called a receding horizon controller
63 The infinite horizon case Elements of P matrix Time step Notice that elt's of P stop changing (much) more than 20 or 30 time steps prior to horizon. what does this imply about the infinite horizon case?
64 The infinite horizon case Elements of P matrix Converging toward fixed P Time step Notice that elt's of P stop changing (much) more than 20 or 30 time steps prior to horizon. what does this imply about the infinite horizon case?
65 The infinite horizon case We can solve for the infinite horizon P exactly: Discrete Time Algebraic Riccati Equation
66 So, what are we optimizing for now? Given: System: Cost function: where: Initial state: Calculate: U that minimizes J(X,U)
67 So, how do we control this thing?
68 Overview 1. expressing a linear system in state space form 2. discrete time linear optimal control (LQR) 3. linearizing around an operating point 4. linear model predictive control 5. LQR variants 6. model predictive control for non-linear systems
69 Inverted pendulum EOM for pendulum: How do we get this system in the standard form:?
70 Inverted pendulum EOM for pendulum: How do we get this system in the standard form:
71 Inverted pendulum EOM for pendulum: How do we get this system in the standard form:!!!!!!!
72 Linearizing a non-linear system Idea: use first-order Taylor series expansion original non-linear system
73 Linearizing a non-linear system Idea: use first-order Taylor series expansion original non-linear system Linearize about first order term
74 Linearizing a non-linear system Idea: use first-order We Taylor just linearized series expansion the system about x^* original non-linear system Linearize about first order term
75 Linearizing a non-linear system Suppose that x^* is a fixed point (or a steady state) of the system... Then:
76 Linearizing a non-linear system Suppose that x^* is a fixed point (or a steady state) of the system... Then: where Change of coordinates
77 Example: pendulum Unit length
78 Example: pendulum Unit length Linearize about:
79 Example: pendulum where
80 Overview 1. expressing a linear system in state space form 2. discrete time linear optimal control (LQR) 3. linearizing around an operating point 4. linear model predictive control 5. LQR variants 6. model predictive control for non-linear systems
81 Linear Model Predictive Control Drawbacks to LQR: hard to encode constraints suppose you have a hard goal constraint? suppose you have piecewise linear state and action constraints? Answer: solve control as a new optimization problem on every time step
82 Linear Model Predictive Control Given: System: Cost function: where: Initial state: Calculate: U that minimizes J(X,U)
83 Linear Model Predictive Control Given: System: Cost function: We're going to solve this problem by expressing it explicitly as where: a quadratic program Initial state: Calculate: U that minimizes J(X,U)
84 Quadratic program Minimize: Subject to:
85 Quadratic program Constants are part of problem statement: Minimize: Subject to: x is the variable Problem: find the value of x that minimizes the objective subject to the constraints
86 Quadratic program Quadratic objective function Minimize: Linear inequality constraints Subject to: Linear equality constraints
87 Quadratic program Minimize: Subject to:
88 Quadratic program Why? Minimize: Subject to:
89 Quadratic objective function Quadratic program
90 Quadratic program Quadratic objective function Inequality constraints
91 Quadratic program Quadratic objective function equality constraints
92 QP versus Unconstrained Optimization Original QP Minimize: Subject to:
93 QP versus Unconstrained Optimization Unconstrained version of original QP Minimize: Subject to:
94 QP versus Unconstrained Optimization Unconstrained version of original QP Minimize: How do we minimize this expression?
95 QP versus Unconstrained Optimization Unconstrained version of original QP Minimize: How do we minimize this expression?
96 Linear Model Predictive Control Minimize: Subject to:
97 Linear Model Predictive Control Minimize: Subject to: What are the variables?
98 Linear Model Predictive Control Minimize: Subject to: What other constraints might we want add?
99 Linear Model Predictive Control Minimize: Subject to:
100 Linear Model Predictive Control Minimize: Subject to: Can't express these constraints in standard LQR
101 Linear MPC Receding Horizon Control Re-solve the quadratic program on each time step: always plan another T time steps into the future Minimize: Subject to:
102 Controllability A system is controllable if it is possible to reach any goal state from any other start state in a finite period of time. When is a linear system controllable? It's property of the system dynamics...
103 Controllability A system is controllable if it is possible to reach any goal state from any other start state in a finite period of time. When is a linear system controllable? Remember this?
104 Controllability What property must this matrix have?
105 Controllability This submatrix must be full rank. i.e. the rank must equal the dimension of the state space
106 NonLinear Model Predictive Control Given: System: Cost function: where: Initial state: Calculate: U that minimizes J(X,U)
107 NonLinear Model Predictive Control Given: System: Cost function: where: Initial state: Calculate: U that minimizes J(X,U)
108 NonLinear Model Predictive Control Given: System: Cost function: where: Initial state: Calculate: U that minimizes J(X,U)
109 NonLinear Model Predictive Control Minimize: Subject to: But, this is a nonlinear constraint so how do we solve it now? Sequential quadratic programming iterative numerical optimization for problems with non-convex objectives or constraints similar to Newton's method, but it incorporates constraints on each step, linearize the constraints about the current iterate implemented by FMINCON in matlab...
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