on the design of first order controllers for unstable ... · design for a class of unstable...

$: On the Design of First Order Controllers for Unstable ... · design for a class of unstable infinite dimensional plants, including systems with time delays, fractional order systems,$
On the Design of First Order Controllers for Unstable Infinite Dimensional Plants

Workshop @ LSS, November 2012

Hitay ÖzbayBilkent University

Ankara, [email protected]

A. Nazlı GündeşUniversity of California

Davis, CA, USA

Özbay and Gündeş Workshop @ LSS, November 2012 2

Outline

• The class of plants considered: examples from applications

• Controller structures: PD, PI, PID and first order lead-lag

• Gain margin optimization with PD/lead-lag controllers

• Optimizing integral action gain in PI and PID controllers

• Conclusions and future directions

Structure of the Plants Considered


: stable

Here, we consider a relatively simple class of SISO plants for thepresentation purpose. For possible extensions to larger classes ofMIMO plants with less restrictive assumptions please see thereferences.

We deal with systems with single unstable pole,factorized as follows:


Example 1a: Integrating systems with transport delay(many application examples; including oil/gas pipelines, communication networks, manufacturing plants, storage systems, etc)

flow sentflow in

flow out

accumulation

transport delay


Example 1b: Abstract model of an aircraft

This is an abstract model of an aircraft, at an unstable operating point, for the purpose of controlling the high frequency longitudinal dynamics (short period). High frequency dynamics due to elasticity, sensor, actuator, sampling, contribute to the time delay. Depending on the operating regime, in X-29 aircraft, we have:(Enns-Ozbay-Tannenbaum, 1992)

X-29 aircraft


Example 2a: Flexible beam with non-collocated actuator and sensor

force input

sensor

Actuator

displacement:

velocity output

from PDEs

Controller

transmission delay


Example 2b: Flexible robot arm with collocated actuator and sensor

Transfer function from torque to angular velocity (via finite element modeling).


Example 3: Interconnected time delay systems


Example 4: A non-laminated magnetic suspension system (a fractional order system); Knospe and Zhu (2011)

Controller Structure


Lead or lag controller

PD-controller

PI-controller


Two Stage Design – to be used for integral action controllers

Assume, then if

Let a controller with integral action be given in the form

Notation: When the feedback system, formed by controller Cand plant P, is stable we say that the controller C stabilizes P. For a given P the set of all stabilizing controllers is denoted by

Stabilizing First Order Controllers


Note that or if and only if

Let and define

Then

where

or

is unimodular for some where


is unimodular if

independent of

A less conservative condition:

Clearly


Example: Allowable gains of the proportional controller

A conservative stability condition:

Exact condition for stability:

A less conservative condition:

next page


From this graph we see that the less conservative bound of the gain is approximately equal to

10-2 10-1 100 101 10210-2

10-1

100

101

102

X: 0.1485Y: 0.1485

a versus 1/|| ||

a

X: 0.3725Y: 0.3705

X: 1.481Y: 1.479

X: 3.69Y: 3.693

X: 14.76Y: 14.78

X: 36.89Y: 36.93

h=10

h=4

h=1

h=0.4

h=0.1

h=0.04

Gain Margin Optimization


The controller in the form

stabilizes the plant for all gain values in the interval

In order to maximize the largest allowable controller gain

we would like to minimize or

by adjusting the parameters


When does this method fail?

We have seen that there is some conservatism in this approach; i.e. there may exist proportial controllers for some plants, that are not possible to detect here.

There are also some plants for which there does not exist a stabilizing proportional controller. Clearly, in this case it is not possible to find a feasible gain parameter a satisfying

Physical restrictions:

Example:

If there exists a ≥ 0 satisfying

then we can find a stabilizing controller using this method.Note that this condition is stronger than p.i.p.


Example:

Parity interlacing property is satisfied if and only if


PD-Controller Case:

Define

See Ozbay and Gundes Appl. Comp. Math. (2007) for a detailed discussion on the computation of the optimal Q, minimizing , for the case a = 0.


Algorithm (GMoptPD):

gain margin optimization when

involves drawing a graph like the one on p.15


Example

10-2 10-1 100 101 102100

101

102

X: 13.22Y: 14.23

Proportional control only:


PD control improves the upper bound on the allowable gain:

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.22

4

6

8

10

12

14

16 X: 0.02Y: 15.23

a max

(Q)

Q

GMoptPD controller:

Least fragile PD controller in this setting:


Lead or Lag Controller Case:

Corresponding gain margin optimization algorithm in Step 1 has to be done for two parameters in nested loops; then in Step 2 we will obtain a surface plot, and find its maximum. This brute force search can be computationally demanding.

Adding Integral Action


Characteristic equation:

Define then

Consider a new controller with additional integral action:

Optimal PI conroller with largest allowable :


if or

Least fragile integral action gain:


Example

10-2 10-1 100 101 102100

101

102

X: 13.22Y: 14.23


2 3 4 5 6 7 8 9 10 11 120

1

2

3

4

5

6

Kp

b max

Kp versus bmax(Kp)

5.7

4.8

Conclusions and Future Directions


We have seen a design method for first order controllerdesign for a class of unstable infinite dimensional plants,including systems with time delays, fractional ordersystems, systems represented by PDEs.

The approach is based on the small gain theorem andrequires minimization of an H∞ norm over a low numberof parameters.

Recently developed optimization methods such as“hifoo/hanso” can be used for the solution of the H∞minimization over a low number of parameters.


Gain margin optimization for the same class of plants using infinite dimensional stable controllers:

Let C0 be an arbitrary order stable controller

For define

All controllers in the form stabilize the plant if

If

then, define

Find

References (in random order)


1. H. Özbay, A. N. Gündeş, “Resilient PI and PD Controller Designs for a Class ofUnstable Plants with I/O Delays,” Appl. Comp. Math., 2007, pp. 18—26.

2. A. N. Gündeş, H. Özbay, A. B. Özgüler, “PID controller synthesis for a class ofunstable MIMO plants with I/O delays”, Automatica, 2007, pp. 135—142.

3. D. Üstebay, H. Özbay, A. N. Gündeş, “A new PI and PID control design methodfor integrating systems with time delays: applications to AQM of TCP flows,”WSEAS Trans. Systems and Control, 2007, pp. 117—124.

4. H. Özbay, C. Bonnet, A. R. Fioravanti, “PID controller design for fractional-ordersystems with time delays”, Systems & Control Letters, 2012, pp. 18–23.

5. A. N. Gündeş and H. Özbay, “Low Order Controller Design for Systems with TimeDelays”, Proc. of the 50th IEEE Conference on Decision and Control, Orlando,Florida, USA, December 2011, pp. 5633—5638.

6. D. Melchor-Aguilar, S.-I. Niculescu, Computing nonfragile PI controllers for delaymodels of TCP/AQM networks, Int. J. Control, 2009, pp. 2249–2259.


7. L-L. Ou, W-D. Zhang, L.Yu, “Low-Order Stabilization of LTI Systems With TimeDelay,” IEEE Trans. Automatic Control, 2009, pp.774-787.

8. S. Gumussoy, W. Michiels, “Fixed-Order H∞ Control for InterconnectedSystems using Delay Differential Algebraic Equations”, SIAM J. Control andOpt., 2011, pp. 2212—2238.

9. G. J. Silva, A. Datta, S. Bhattacharyya, PID Controllers for Time-DelaySystems, Birkhauser, 2004.

10.A. R. Fioravanti, C. Bonnet, H. Özbay, S-I. Niculescu, “A numerical method forstability windows and unstable root-locus calculation for linear fractional time-delay systems”, Automatica, (2012) DOI: 10.1016/j.automatica.2012.04.009.

11.W. Michiels, S. I. Niculescu, Stability and Stabilization of Time-Delay Systems:An Eigenvalue-Based Approach, SIAM, 2007.

12.H. Özbay, “Robust Control of Infinite Dimensional Systems: Theory andApplications,” Proc. of the 17th International Symposium on MathematicalTheory of Networks and Systems (MTNS 2006), Kyoto, Japan, July 2006

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