Ch_ 3

3.11 Problems 323

Let Z be the matrix

A -BR;~B~

3-642

-R1 -AT

Z can always he represented as

where J is the Jordan canonical form of Z. I t is always possible to arrange the columns of W such that J can be partitioned as

Here J,,, J,, and J,, are 11 x 11 blocks. Partition W accordingly as

and show by considering the 12and 22-blocks of this equality that if W12 is nonsingular P = WzCK: is a solution of the algebraic Riccati equation. Note that in this manner many solutions can be obtained by permuting the

3.9*. Steady-state solrrtion of the Riccati eqtiation by clingo~~alization

Consider the 2n x 2n matrix Z as given by 3-247 and suppose that it cannot be diagonalized. Then Z can be represented as

where J i s the Jordan canonical form ofZ, and Wis composed of the characteristic vectors and generalized characteristic vectors of Z. I t is always possible to arrange the columns of W such that J can be partitioned as follows

where the n x 11 matrix J, has as diagonal elements those characteristic values of Z that have positive real parts and half of those that have zero

324 Optimal Linear State flcedbaclc Conhol Systems

real parts. Partition Wand V = W-I accordingly as

Assume that {A, B) is stahilizable and {A,D)detectable. Follow the argument of Section 3.4.4 closely and show that for the present case the following conclusions hold.

(b) W13 is nonsingular and

P= w,,wz.

(c)The steady-state optimal behavior of the state is given by

Hence Z has no characteristic values with zero real parts, and the steadystate closed-loop poles consist of those cbaracterstic values of Z that have negative real parts. Hint: Show that

where the precise form of X(t) is unimportant.

3.10*. Bass' relationfor P (Bass, 1967)

Consider the algebraic Riccati equation

and suppose that the conditions are satisfied under which it has a unique nonnegative-definite symmetric solution. Let the matrix Z be given by

It follows from Theorem 3.8 (Section 3.4.4) that Z has no characteristic values with zero real parts. Factor the characteristic polynomial of Z as follows

satisfies the relation:

Hint: Write $(Z)= $(WJW-I) = W $ ( aW-1 = W$(.7)lT where V = W-I and J = diag (A,- A ) in the notation of Section 3.4.4.

3.11'. Negative espo~~entiolso111tionof the Riccati eqlrotiorl (Vaughan,

1969)

using the notation of Section 3.4.4, show that the solution of the timeinvariant Riccati equation

can be expressed as follows:

Show with the aid of Problem 3.12 that S can also be written in terms of Was

3.12*. The re loti or^ between Wand V

Consider the matrix Z as defined in Section 3.4.4.

(a) Show that if e = col (e', e"), where e' and e" both are n-dimensional vectors, is a right characteristic vector o f 2 corresponding to the characteristic value 1, that is, Ze = Ae, then (e"', -elT) is a left characteristic vector o f Z corresponding to the characteristic value -1, that is,

(b) Assume for simplicity that all characteristic values At, i = 1, 2 , ...,211, of Z are distinct and let the corresponding characteristic vectors be given by e$.9 i = 1,2, ...,2n. Scale the e j such that if the characteristic vector e = col (e', e'') corresponds to a characteristic value 1, and f = col (7,f " ) corresponds to -A, then

Show that if W is a matrix of which the columns are e,, i = 1,2: ...,211, and we partition

326 Optimnl Linear Stnte Feedback Control Systems

then (O'Donnell, 1966; Walter, 1970)

Hint: Remember that left and right characteristic vectors for different characteristic values are orthogonal.

3.13'. Freqltericy do~iiainsol~ttionof regdator problenis

For single-input time-invariant systems in phase-variable canonical form, the regulator problem can be conveniently solved in the frequency domain. Let

3-668

be given in phase-variable canonical form and consider the problem of minimizing

(a) Show that the closed-loop characteristic polynomial can be found by

3-671

P

where H(s) is the open-loop transfer function H(s) = d(sI - A)-'b.

(b) For a given closed-loop characteristic polynomial, show how the corresponding control law

(a) Show that as p L O some of the closed-loop poles go to infinity while the others stay finite. Show that those poles that remain finite approach the left-half plane zeroes of

number of zeroes of 3-675.Compare the proof of Theorem 1.19 (Section

Suppose that a Bode plot is available of the open-loop frequency response function H@) = d ( j o I - A)-lb. Show that for small p the radius of the

faraway poles of the steady-state optimal closed-loop system canbeestimated:

-

as the frequency w, for which IH(jw.)I = JP.