Optimal Control

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# Optimal Control - PowerPoint PPT Presentation

Optimal Control. Motivation Bellman’s Principle of Optimality Discrete-Time Systems Continuous-Time Systems Steady-State Infinite Horizon Optimal Control Illustrative Examples. Motivation. Control design based on pole-placement often has non unique solutions

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Presentation Transcript
Optimal Control
• Motivation
• Bellman’s Principle of Optimality
• Discrete-Time Systems
• Continuous-Time Systems
• Steady-State Infinite Horizon Optimal Control
• Illustrative Examples
Motivation
• Control design based on pole-placement often has non unique solutions
• Best locations for eigenvalues are difficult to determine
• Optimal control minimizes a performance index based on time response
• Control gains result from solving the optimal control problem

Where Q is a symmetric (QT=Q) nxn matrix and b is an nx1 vector

It can be shown that the Jacobian of f is

Matrix representation:

The value of x that minimizes f(x) (denoted by x*) sets

or equivalently

Provided that the Hessian of f,

is positive definite

Positive Definite Matrixes

Definition: A symmetric matrix H is said to be positive definite (denoted by H>0) if xTHx>0 for any non zero vector x (semi positive definite if it only satisfies xTHx0 for any x (denoted by H  0) ).

Positive definiteness (Sylvester) test: H is positive definite iff all the principal minors of H are positive:

Optimal solution:

Thus x* minimizes f(x)

Discrete-Time Linear Quadratic (LQ) Optimal Control

Given discrete-time state equation

Find control sequence u(k) to minimize

Comments on Discrete-Time LQ Performance Index (PI)
• Control objective is to make x small by penalizing large inputs and states
• PI makes a compromise between performance and control action

uTRu

xTQx

t

Principle of Optimality

4

2

7

5

9

1

3

8

6

Bellman’s Principle of Optimality: At any intermediate state xi in an optimal path from x0 to xf, the policy from xi to goal xf must itself constitute optimal policy

Discrete-Time LQ Formulation

Optimization of the last input (k=N-1):

Where x(N)=Gx(n-1)+Hu(n-1). The optimal input at the last step is obtained by setting

Solving for u(N-1) gives

Optimal Value of JN-1

Substituting the optimal value of u(N-1) in JN gives

The optimal value of u(N-2) may be obtained by minimizing

where

But JN-1 is of the same form as JN with the indexes decremented by one.

Summary of Discrete-Time LQ Solution

Control law:

Ricatti Equation

Optimal Cost:

• Control law is a time varying state feedback law
• Matrix Pk can be computed recursively from the Ricatti equation by decrementing the index k from N to 0.
• In most cases P and K have steady-state solutions as N approaches infinity
Matlab Example

y

Find the discrete-time (T=0.1) optimal controller that minimizes

u

M=1

Solution: State Equation

Discretized Equation and PI Weighting Matrices

Discretized Equation:

Performance Index Weigthing Matrices:

R

Q

PN

System Definition in Matlab

%System: dx1/dt=x2, dx2/dt=u

%System Matrices

Ac=[0 1;0 0]; Bc=[0;1];

[G,H]=c2d(Ac,Bc,0.1);

%Performance Index Matrices

N=100;

PN=[10 0;0 0]; Q=[1 0;0 0]; R=2;

Ricatti Equation Computation

%Initialize gain K and S matrices

P=zeros(2,2,N+1); K=zeros(N,2);

P(:,:,N+1)=PN;

%Computation of gain K and S matrices

for k=N:-1:1

Pkp1=P(:,:,k+1);

Kk=(R+H'*Pkp1*H)\(H'*Pkp1*G);

Gcl=G-H*Kk;

Pk=Gcl'*Pkp1*Gcl+Q+Kk'*R*Kk;

K(k,:)=Kk; P(:,:,k)=Pk;

end

LQ Controller Simulation

%Simulation

x=zeros(N+1,2); x0=[1;0];

x(1,:)=x0';

for k=1:N

xk=x(k,:)';

uk=-K(k,:)*xk;

xkp1=G*xk+H*uk;

x(k+1,:)=xkp1';

end

%plot results

Given discrete-time state equation

Find control sequence u(k) to minimize

Solution is obtained as the limiting case of Ricatti Eq.

Summary of LQR Solution

Control law:

Ricatti Equation

Optimal Cost: