Advanced Control Engineering - Chapter 9 pptx

9.1.1 Types of optimal control problems a The terminal control problem: This is used to bring the system as close as possible to a given terminal state within a given period of time.. e

Optimal and robust control

system design 9.1 Review of optimal control

An optimal control system seeks to maximize the return from a system for theminimum cost In general terms, the optimal control problem is to find a control uwhich causes the system

differ-An alternative procedure is the dynamic programming method of Bellman (1957)which is based on the principle of optimality and the imbedding approach Theprinciple of optimality yields the Hamilton±Jacobi partial differential equation,whose solution results in an optimal control policy Euler±Lagrange and Pontrya-gin's equations are applicable to systems with non-linear, time-varying state equa-tions and non-quadratic, time varying performance criteria The Hamilton±Jacobiequation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index),which takes the form of the matrix Riccati (1724) equation This produces an optimalcontrol law as a linear function of the state vector components which is always stable,providing the system is controllable

9.1.1 Types of optimal control problems

(a) The terminal control problem: This is used to bring the system as close as possible

to a given terminal state within a given period of time An example is an

automatic aircraft landing system, whereby the optimum control policy will focus

on minimizing errors in the state vector at the point of landing

(b) The minimum-time control problem: This is used to reach the terminal state in the

shortest possible time period This usually results in a `bang±bang' control policywhereby the control is set to umaxinitially, switching to uminat some specific time

In the case of a car journey, this is the equivalent of the driver keeping his footflat down on the accelerator for the entire journey, except at the terminal point,when he brakes as hard as possible

(c) The minimum energy control problem: This is used to transfer the system from an

initial state to a final state with minimum expenditure of control energy Used insatellite control

(d) The regulator control problem: With the system initially displaced from

equilib-rium, will return the system to the equilibrium state in such a manner so as tominimize a given performance index

(e) The tracking control problem: This is used to cause the state of a system to track

as close as possible some desired state time history in such a manner so as tominimize a given performance index This is the generalization of the regulatorcontrol problem

9.1.2 Selection of performance index

The decision on the type of performance index to be selected depends upon the

nature of the control problem Consider the design of an autopilot for a racing yacth

Conventionally, the autopilot is designed for course-keeping, that is to minimisethe error e(t) between that desired course d(t) and the actual course a(t) in the

presence of disturbances (wind, waves and current) Since d(t) is fixed for most of

the time, this is in essence a regulator problem

Using classical design techniques, the autopilot will be tuned to return the vessel onthe desired course within the minimum transient period With an optimal control

strategy, a wider view is taken The objective is to win the race, which means

completing it in the shortest possible time This in turn requires:

(a) Minimizing the distance off-track, or cross-track error ye(t) Wandering off track

will increase distance travelled and hence time taken

(b) Minimizing course or heading error e(t) It is possible of course to have zero

heading error but still be off-track

(c) Minimizing rudder activity, i.e actual rudder angle (as distinct from desired

rudder angle) a(t), and hence minimizing the expenditure of control energy

(d) Minimizing forward speed loss ue(t) As the vessel yaws as a result of correcting

a track or heading error, there is an increased angle of attack of the total velocityvector, which results in increased drag and therefore increased forward speedloss

From equation (9.2) a general performance index could be written

J ˆ

Z t1

t h( ye(t), e(t), ue(t), a(t))dt (9:3)

Quadratic performance indices

If, in the racing yacht example, the following state and control variables are defined

x1ˆ ye(t), x2ˆ e(t), x3ˆ ue(t), u ˆ a(t)then the performance index could be expressed

J ˆ

Z t1

t 0f(q11x1‡ q22x2‡ q33x3) ‡ (r1u)gdt (9:4)or

3

5 xx12

x3

24

3

5 ‡ [u][r1][u]

24

35dt

where over the time interval t0to t1,

f (x, t0) ˆ f (x(0))

f (x, t1) ˆ 0From equations (9.1) and (9.2), a Hamilton±Jacobi equation may be expressed as

@tx ˆ minu xTQx ‡ uTRu ‡ 2xTP(Ax ‡ Bu) (9:17)

To minimize u, from equation (9.17)

@[@f /@t]

Equation (9.18) can be re-arranged to give the optimal control law

uoptˆ Kx (9:20)where

Equation (9.23) belongs to a class of non-linear differential equations known as thematrix Riccati equations The coefficients of P(t) are found by integration in reversetime starting with the boundary condition

Kalman demonstrated that as integration in reverse time proceeds, the solutions ofP(t) converge to constant values Should t1 be infinite, or far removed from t0, thematrix Riccati equations reduce to a set of simultaneous equations

The discrete solution of the matrix Riccati equation solves recursively for K and P inreverse time, commencing at the terminal time, where

K(N (k ‡ 1)) ˆ [TR ‡ BT(T)P(N k)B(T)] 1BT(T)P(N k)A(T) (9:29)

P(N (k ‡ 1)) ˆ [TQ ‡ KT(N (k ‡ 1))TRK(N (k ‡ 1))] ‡ [A(T)

B(T)K(N (k ‡ 1))]TP(N k)[A(T) B(T)K(N (k ‡ 1))]

(9:30)

As k is increased from 0 to N 1, the algorithm proceeds in reverse time When run

in forward-time, the optimal control at step k is

Example 9.1 (See also Appendix 1, examp91.m)

The regulator shown in Figure 9.1 contains a plant that is described by

_x1_x2

y ˆ [1 0]xand has a performance index

(a) the Riccati matrix P

(b) the state feedback matrix K

(c) the closed-loop eigenvalues

" #1[ 0 1 ] p11 p12

p12 ˆ p21ˆ 0:732 and 2:732Using positive value

p22 ˆ 0:542 and 4:542Using positive value

(9:45)(b) Equation (9.21) gives the state feedback matrix

K ˆ R 1BTP ˆ 1[ 0 1 ] 2:403 0:732

0:732 0:542

(9:46)Hence

K ˆ [ 0:732 0:542 ]

ˆ 0

... integrators in thecontroller, steady-state errors must be expected However, the selection of theelements in the Q matrix, equation (9. 90), focuses the control effort on control-

0... uT(k)Ru(k)]T (9: 52)

Discrete minimization gives the recursive Riccati equations (9. 29) and (9. 30) These

are run in reverse-time together with the discrete reverse-time state tracking... class="text_page_counter">Trang 12

The reverse-time calculations are shown in Figure 9. 3 Using equations (9. 29) and

(9. 30) and commencing with P(N) ˆ 0, it can be