dft, Feedback Control Theory, Doyle

Chapter 5, Stabilization

Coprime Factorization

Controller Parametrization

Strong and Simultaneous Stabilization

Say that a plant is strongly stabilizable if internal stabilization can be achieved with C itself a stable transfer function.

Theorem 3 P is strongly stabilizable iff it has an even number of real poles between every pair of real zeros in Res ≥ 0.

Two plants P1 and P2 are simultaneously stabilizable if internal stability is achievable for both by a common controller.

Theorem 4 P1 and P2 are simultaneously stabilizable iff P is strongly stabilizable.

Chapter 6, Design Constraints

Algebraic Constraints

The identity S + T = 1 always holds. This is an immediate consequence of the definitions of S and T. So in particular, |S(jω)| and |T(jω)| cannot both be less than 1/2 at the same frequency ω.

Robust performance ||W1S| + |W2T||∞ < 1 implies min{|W1(jω)|, |W2(jω)|} < 1, ∀ω

If p is a pole of L in Res ≥ 0 and z is a zero of L in the same half-plane, then S(p) = 0, S(z) = 1, T(p) = 1, T(z) = 0.

Analytic Constraints

Bounds on the Weights W1 and W2

|W1S|∞ ≥ |W1(z)| for z a zero of L, S(z) = 1

Performance criterion |W1S|∞ < 1 implies |W1(z)| < 1

|W2T|∞ ≥ |W2(p)| for p a pole of L, T(p) = 1

robust stability criterion |W2T|∞ < 1 implies |W2(p)| < 1

All-Pass and Minimum-Phase Transfer Functions Decomposition

If there are a pole and zero close to each other in the right half-plane, they can greatly amplify the effect that either would have alone.

The Waterbed Effect

Theorem 1 Suppose that P has a zero at z with Rez > 0. Then there exist positive constants c1 and c2, depending only on ω1, ω2, and z, such that c1 logM1 + c2 logM2 ≥ log |Sap(z)^{−1}| ≥ 0.

The waterbed effect is amplified if the plant has a pole and a zero close together in the right half-plane.

The waterbed effect applies to non-minimum-phase plants only. In fact, a very good result in minimum-phase case can be proved (Section 10.1)

Theorem 2 Assume that the relative degree of L is at least 2. Then the area under log |S(jω)| on [0, ∞] is π(log e)( \Sigma Rep_i), where {pi} denote the set of poles of L in Res > 0.

So the negative area, required for good tracking over some frequency range, must unavoidably be accompanied by some positive area

The waterbed effect applies to non-minimum-phase systems, whereas the area formula applies in general (except for the relative degree assumption).

Chapter 2, Norms for Signals and Systems

G(s) is called stable if it is analytic in the closed right half-plane

G(s) is called proper if G(j∞) is finite, strictly proper if G(j∞) = 0

Chapter 3, Basic Concepts

well-posedness

all closedloop transfer functions exist,

that is, all transfer functions from the exogenous inputs to all internal signals and the outputs of the summing junctions

or stronger: proper and rational iff PCF(∞) != −1

Internal Stability

If the closedloop transfer functions above are stable, then the feedback system is said to be internally stable.

Theorem 1 The feedback system is internally stable iff there are no closed-loop poles in Res ≥ 0.

The closed-loop poles are the zeros of the characteristic polynomial NPNCNF + MPMCMF

Theorem 2 The feedback system is internally stable iff the following two conditions hold:

(a) The transfer function 1 + PCF has no zeros in Res ≥ 0.

(b) There is no pole-zero cancellation in Res ≥ 0 when the product PCF is formed.

Performance

sensitivity function S := 1 / (1 + L), e = Sr

Performance specification |W1S|∞ < 1

Why we use ∞-norm bound on W1S

Graphical interpretation

the complementary sensitivity function T := 1 − S

Conclusion: Performance specs that involve e result in weights on S and performance specs on u result in weights on T.

Chapter 4, Uncertainty and Robustness

Plant Uncertainty

(allowable) Multiplicative Perturbation model

allowable: no unstable poles of P are canceled in forming ~{P}.

Feedback Uncertainty model

Robust Stability

A controller C provides robust stability if it provides internal stability for every plant in P a set of uncertainty model.

Theorem 1 (Multiplicative uncertainty model) C provides robust stability iff |W2T|∞ < 1.

The key equation is $1 + (1 + \DeltaW2)L = (1 + L)(1 + \DeltaW2T)$

Graphical interpretation

Robust Performance

Robust performance condition is a combination of robust stability condition and nominal performance condition (for all plant in P), i.e., |W2T|∞ < 1 and |W1~{S}|∞ < 1 (in multiplicative uncertainty model)

Theorem 2 A equivalent condition for robust performance is ||W1S| + |W2T||∞ < 1.
(Multiplicative uncertainty model)

Graphical interpretation