Multivariable Issues - Directionality

Output

6.2.4 Multivariable Issues - Directionality

In Section 2.2 we outlined a general windup compensation scheme which avoids windup in the states of K ( s). For SISO plants this approach leads to graceful per- formance degradation when the system enters the nonlinear operating regime as a result of saturations. While our state space approach allows us to extend the windup compensation scheme to MIMO plants in a natural manner, windup compensation alone, is often not adequate to ensure graceful performance degradation in the MIMO case. We demonstrate this with a simple example.

Example 2

We consider the 2 x 2 plant,

( 6.32)

with both inputs limited by ±15.0. The controller, selected on the basis of linear performance is,

J((s)=

10s+l [4 5]

8 3 4

The linear response shown in Figure 7a for a setpoint change, r

= [:~i],

^is

decoupled with a first order response in each output with time constant of 1.0 and no overshoot. The nonlinear response to this same setpoint change is shown in Figure 76.

Both outputs overshoot significantly {approximately 500% at t

=

⁴⁾ then overcorrect and m11-lershoot (approximately 100% at t

=

8) before settling.

With a broad definition this drastic performance deterioration would be assigned to "windup problems." In fact only the smaller undershoot problem is the result of windup. This can be seen in Figure 7c where we have included windup compensation, (6.27), in the nonlinear simulation. The large initial overshoot is still present and the smaller undershoot (around t

=

8) ha..i;; b~n eliminate,L It. is clPa.r then that relative to the large initial overshoot, windup is a relatively minor problem in this example.

The problem demonstrated in the preceding example is unique to MIMO systems and results from the directional nature of the plant. In MIMO systems the plant gain is a function of the input direction. Since the saturation operates element by element on u to generate

u

the direction of

u

is different than that of u. For example if u

= [i:~],

the resulting ft is ft

= [i:~]

and the direction of the controller output.

u, is is different than the plant input,

u.

^If the saturation error, u -

u,

corresponds to the high plant gain direction, the difference in plant outputs corresponding to u and

u

will be maximal. Correspondingly if u -

u

is aligned with the low plant gain direction, the effect on the output will be relatively modest.

Outputs Inputs

1--- 70-r---

o.e o.,

0.4

0.2

2 l

50 30

10 1

O-t---1 -10-+---

2

5 3

1 -1

-3 -5

0 10 20 30 0 10 20 30

Figure 7a.: Example 2 - Step response for the unconstrained :sy:slem.

Outputs Inputs

20 15

2 10

1

5 2

0

0 10 20 30 0 10 20 30

Figure 7b: Example 2 - Step response for the constrained system with no saturation compensation.

Outputs Inputs

.5 20

3 1.5

1

10

-1 l

-3 ⁵ 2

-5 0

0 10 20 30 0 10 20 30

Figure 7c: Example 2 - Step response for the constrained system with saturation compensation (6.27).

Outputs

1 - - - -

0.8 0.6

0.2

1

2

0 - + - - - 1

0 10 20 30

20 15 10

s

0

Inputs

- l

^l

..

2

I I

0 10 20 30

Figure 7d: Example 2 - Step response for the constrained system with saturation compensation (6.27) and directionality compensation (6.34).

d

r _K ^y

R

Figure 8: The feedback system with saturation compensation, ^R, and directionality compensation, R_{2 •}

Since the saturation operator acts element by element on u its structure is di- agonal. It is well-known from robust linear control theory that some plant and controller combinations experience severe performance det.eriora.tion in the prPsPnre of diagonal input uncertainties. Specifically, ill-conditioned systems (those having large scaled condition numbers) together with mverse based ( and consequently ill- conditioned) compensators as in the current example, have this property (see for example (87]). Loosely speaking the diagonal operator disturbs the inversion so that P(.s)sF1.tK(s) ~ L(s) though P(s)K(s) ~ L(s), where L(s) is the desired loopshape.

\Ve can eliminate the directionality problem by adjusting all of the elements in u when one of them becomes saturated so that u and

u

have the same direction ( a similar approach was adopted in [35l). This can be achieved by inserting an additional block in the loop as in Figure 8. Here the block R₂ is a nonlinear operator described by

u'

-

^{.H2u (6.3{)}

{ 11•;~

llulloo :S

1

(G.:J,})

- llulloo >

l

where

llulloo =

max

luil

I

(6.:36) The purpose of R2 is to scale back the controller output until its largest element has magnitude one. In this case the saturation will have no effect since its input,

u',

always has

lui I :5

1 for all i. What we have effectively done then is replace the diagonal saturation operator by a scalar times identity operator. In this case, if we allow d₂ to be the scalar valued describing function appropriate for the composite operator, satR_{2 ,} we have

(6.37)

and we see that ( to a first approximation) the desired loop shape is only perturbed by a scalar facto,r. The impact on the closed loop is now not dependent on the direction of u but only on its magnitude.

Using this approach to directionality compensation we return to Example :2 and simulate the response to the same setpoint change, r

= [:1i].

The response, shown in Figure 7d, is well behaved with no over or undershoot characteristic of windup or directionality problems.

It should be noted that this approach to directionality compensation is not neces- sarily optimal. Indeed it may happen that without directionality compensation u -

u

is in the low plant gain direction. If information is available regarding the directional characteristics of the plant a constrained optimization can be performed to find a u' which minimizes the input of the saturation on the output error ( e.g., minimizing the component of u. - u.' in the high gain plant direction). These schemes are typically very complicated and computationally intensive. We favor this simple scheme because it is insensitive to the directionality of the plant (and hence requires no such information), has provided very good results, is amenable to available analysis techniques, and is trivial to implement. Implicit in our assumption that I< ( s) is designed appropriately for the linear plant is the assumption that the output of K(s), u, is in the appropriate direction. \Vith this in mind the simple directionality approach seems justified.