Infinite Time LQR (LTV)System
System:
PI:
Note that in this case, (4-29) must be completely controllable.
Since, otherwise The PI will be infinite.
Optimal Control :
where
(4-29)
(4-30)
(4-31)
2
Infinite Time LQR (LTV )System
DRE:
Final Condition:
Optimal Cost:
(4-32)
(4-33)
Infinite Time LQR (LTV) System
4
Plant:
PI:
System (4-35) must be controllable. Since, otherwise PI will be infinite. That is, the following matrix must be nonsingular.
In this case, it can be shown that DRE is converted to a
nonlinear, matrix
, algebraic Riccati equation (ARE).
Infinite time LQR (LTI) System
(4-35)
(4-36)
(4-37)
Infinite time LQR (LTI) System
6
Implementation of the Closed-Loop Optimal Control: Infinite
Final Time
Example
8
Example
Example
Note that matrix must be positive semi-definite. P
10
Example
Example
12
Example
Example
14
Analytical Solution of the Algebraic Riccati Equation
As with finite time case, the analytical solution of ARE is obtained as follows:
Some points on Stability Condition
1. To guarantee the stability of the recent closed-loop optimal control system, it is necessary the pair [A,C] is detectable, where C is any matrix such that CTC = Q.
2. The Riccati coefficient matrix is positive definite if and only if [A, C] is completely observable.
3. Thus both detectability and stabilizability conditions are necessary for the existence of a stable closed-loop system.
P
16
Linear Quadratic Tracking (LQT) Systems
Plant:
Our objective is to control the system (4.38) in such a way that the output y(t) tracks the desired output z(t) as close as possible during the interval [t0, tf] with minimum expenditure of control effort.
Define error vector :
PI:
t f is fixed and x(tf) is free.
(4-38)
(4-39)
Solution to the LQT
optimal control:
(4-40)
18
Matrices:
Optimal State (trajectories) is the solution of
Optimal Cost
Solution to the LQT
(4-45)
Implementation of the Optimal Tracking System
20
Example
Second order plant:
PI:
The final time tf =20,
final state x(tf) is free,
It is required to keep the state x1(t) close to 1.
Obtain the optimal feedback control
(0) [ 0.5 0.5] ;T x
Example: Solution
P(t) , g(t):
Optimal Control:
DRE:
22
g(t) is the solution of:
Example: Solution
Example: Riccati Coefficients
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
p11
p12 p22
24
Example: Coefficients g 1 , g 2
Example: Optimal States
26
Example: Optimal Control
Exercise
solve the previous example with the following PI
Suppose initial condition: x(0)=[-1 0]T
28
LQT System: LTI & Infinite Time Case
Plant:
Define error vector :
PI:
(4-49)
(4-50)
(4-51)
LQT System: LTI & Infinite Time Case
1
0
T T T
PA A P PBR B P
C QC
ARE:
Vector Function g(t):
( )
T ( ) ( ) g t PE A g t Wz t
1 T , T
E BR B W C Q
where
Optimal Control:
(4-52)
(4-53)
(4-54)
(4-55)
f 0
g t
