最优控制 经典ppt

Department of Automation School of Information Science & Engineering Central South University Changsha, Hunan, 410083, China
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Contents
Chapter 1 Introduction
According to the principle of optimality, if the N -stage decision VN [ x (0)] is optimal,
then the ( N − 1)-stage decision VN −1 [ x(1) ] , regarding the x(1) resulting from x(0)
Recurrently solving from final state:
V (F ) = 0
⎧V (a3 ) = 4 ⎪ ⎨V (b3 ) = 6 ⎪V (c ) = 8 ⎩ 3
⎧V (a2 ) = min { L ( a2 → V (a3 ) ) , L ( a2 → V (b3 ) ) , L ( a2 → V (c3 ) )} = 10 ⎪ ⎪ ⎨V (b2 ) = min { L ( b2 → V (a3 ) ) , L ( b2 → V (b3 ) ) , L ( b2 → V (c3 ) )} = 9 ⎪ ⎪V (c2 ) = min { L ( c2 → V (a3 ) ) , L ( c2 → V (b3 ) ) , L ( c2 → V (c3 ) )} = 8 ⎩
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V ( S ) = min {L ( S → V ( a1 ) ) , L ( S → V (b1 ) ) , L ( S → V ( c1 ) )} = 12
⎧V ( a1 ) = min {L ( a1 → V ( a2 ) ) , L ( a1 → V (b2 ) ) , L ( a1 → V ( c2 ) )} = 9 ⎪ ⎪ ⎨V (b1 ) = min {L ( b1 → V ( a2 ) ) , L ( b1 → V (b2 ) ) , L ( b1 → V ( c2 ) )} = 12 ⎪ ⎪V ( c1 ) = min {L ( c1 → V ( a2 ) ) , L ( c1 → V (b2 ) ) , L ( c1 → V ( c2 ) )} = 11 ⎩
N −1
x(0) = x0 , x( N ) = xN
J N = ∑ L [ x(k ), u (k ), k ] , J,L∈ R
k =0
Define VN − j [ x( j ) ] as the ( N − j )-stage optimal decision starting from state x( j ) :
6.2 Dynamic programming for discrete systems
6.3 Dynamic programming for continuous systems
6.4 Dynamic programming, the minimum principle and the calculus
the optimal control sequence is {u (0)* , u (1)* ,
and the state x ( k )* ( k < N ) is resulted from x (0), u (0)* , u(1)* ,
then, , u( N − 1)* }.
the (N − k )-stage decision problem starting at the state x ( k )* is also optimal, and
k = 0,1, , N −1
Let the state equation of discrete system be:
x(k + 1) = f [ x(k ), u (k ), k ] , x, f ∈ R n , u ∈ R m ,
Solve for optimal control sequence {u (k )* } making J minimal.
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Chapter 6
Dynamic Programming
6.1 Multistage decision process and the principle of optimality
6.2 Dynamic programming for discrete systems
6.3 Dynamic programming for continuous systems
Chapter 7 Minimum Time Optimal Control
Chapter 8 Linear-Quadratic Optimal Control
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Chapter 6
Dynamic ng
6.1 Multistage decision process and the principle of optimality
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— The dynamic programming algorithm, the recurrence formula
Example: The state equation of discrete system is: xi +1 = xi + ui , i = 0,1, , N − 1
VN − j [ x( j ) ] = min { L [ x( j ), u ( j ), j ] + VN − ( j +1) [ x( j + 1) ]}
u(N − j)
u( j)
or
V j [ x( N − j ) ] = min { L [ x( N − j ), u ( N − j ), N − j ] + V j −1 [ x( N − j + 1) ]}
and u (0) as the initial state, should be also optimal:
N −1 ⎧ ⎫ ∴ VN [ x(0) ] = min ⎨ L [ x(0), u (0), 0] + min ∑ L [ x(k ), u (k ), k ]⎬ u (0) u (1), u (2), , u ( N −1) k =1 ⎩ ⎭
6.1 Multistage decision process and the principle of optimality
V[x(k)]: the optimal decision at the state x(k); a shortest path from the state x(k) to terminal F.
* N− j
VN − j [ x( j ) ] = J
⎧ N −1 ⎫ N −1 = min ⎨ ∑ L [ x(k ), u (k ), k ]⎬ = ∑ L ⎡ x(k )* , u (k )* , k ⎤ ⎣ ⎦ {u ( k )} k = j ⎩ ⎭ k= j
∴
⎧ N −1 ⎫ J = VN [ x(0) ] = min ⎨∑ L [ x(k ), u ( k ), k ]⎬ {u ( k )} ⎩ k = 0 ⎭
N −1 ⎧ ⎫ = min L [ x(0), u (0), 0] + ∑ L [ x (k ), u (k ), k ]⎬ ⎨ u (0), u (1), , u ( N −1) k =1 ⎩ ⎭ 10
* N
6.2 Dynamic programming for discrete systems
Chapter 2 Static Optimization
Chapter 3 Variational Methods
Chapter 4 The Pontryagin Minimum Principle
Chapter 5 Discrete-Time Optimal Control
Chapter 6 Dynamic Programming
= min { L [ x(0), u (0), 0] + VN −1 [ x(1) ]}
u (0)
Similarly, we have
VN −1 [ x(1) ] = min { L [ x(1), u (1),1] + VN − 2 [ x(2) ]}
u (1)
In general, it holds that
Volume 3: Chapter 6, 7, 8
Modern Control Theory
— Optimal Control —
( An undergraduate course )
Hui PENG
PhD, Professor
( /staffmember/HuiPeng.htm )
6.4 Dynamic programming, the minimum principle and the calculus
of variations
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6.2 Dynamic programming for discrete systems
6.2 Dynamic programming for discrete systems
6.1 Multistage decision process and the principle of optimality
The principle of optimality:
An optimal policy has the property that, whatever the initial state and decision (i.e., control) are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.
of variations
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6.1 Multistage decision process and the principle of optimality
6.1 Multistage decision process and the principle of optimality
Consider the following Shortest Path problem, a multistage decision problem: finding a shortest path from the starting city "S" to the final city "F".
the optimal control of the (N − k )-stage decision problem is {u( k )* ,
One part of optimal trajectory is still optimal trajectory.
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Solve the shortest path problem:
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6.1 Multistage decision process and the principle of optimality
In other words, the principle of optimality states that:
If a N -stage decision problem starting at state x (0) is optimal, , u ( k )* , , u( N − 1)* }, , u( k − 1)* ,
Stage 1 Stage 2 Stage 3
Decision variable: x(1) V[x(1)]
u(1)
u(2) x(2) V[x(2)]
u(3) x(3) V[x(3)]
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State variable: S
F V[F]
Optimal decision: V[S]
V[x(k)]: the optimal decision at the state x(k); a shortest path from the state x(k) to terminal F.