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New Approaches to the Design of Fixed Order Controllers

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### New Approaches to the Design of Fixed Order Controllers

S. P. Bhattacharyya

Department of Electrical Engineering

Texas A & M University

College Station, TX 77843-3128

- New algorithms that can be used to generate the entire set of stabilizing PID and first order controllers for single-input single-output 1) continuous-time rational plants of arbitrary order, 2) discrete-time rational plants of arbitrary order, and 3) continuous-time plants with time delay. A new linear programming algorithm for higher order controllers.
- These algorithms follow from substantial fundamental theoretical advances on PID, first order and arbitrary fixed order stabilization that have been reported by us in recent years. They display the rich mathematical structure underlying the topology of stabilizing sets.
- Examples are presented to clarify the steps involved in implementing the different algorithms.
- The results significantly complement the current techniques for industrial PID design, many of which are adhoc in nature, and open the door to design with fixed order controllers.
- Our solution results in graphical displays of feasible design regions using 2-D and 3-D graphics-should appeal to control designers and are very suitable for computer aided design where several performance objectives have to be overlaid and intersected.

- Fixed order and in particular PID Controllers are widely used in aerospace, motion control, process industries, manufacturing, robotics, disc drives, pneumatic, hydraulic, electrical and mechanical systems
- Current designs are carried out using adhoc tuning rules. These rules have been developed over the years based primarily on empirical observations and industrial experience.
- Modern and Postmodern control theory, that is, state feedback observer based theory of modern and post-modern control theory including optimal control cannot be applied to fixed order controllers. Indeed, until the results to be described here appeared, it was not known how to even determine whether stabilization of a nominal system was possible using PID controllers.
- Our results are first steps toward computer aided designs of fixed order and PID controllers with guaranteed stability, performance and robustness.

Characteristics of PID Controllers

- Provides set point regulation (error zeroing) against arbitrary disturbances (as long as they are low frequency)
- Is robust against modelling errors
- Is nonfragile in general
- Three term controllers are easier to adjust at the design stage as well as online

- Extension of the classical Hermite Bieler Theorem to root counting
- A new Stabilization Algorithm based on this, that generates the entire set of controllers attaining stability
- Calculations based on Linear Programming
- A new Performance Algorithm for Robustness, Gain-Phase Margin and other performance measures generating the entire set attaining specs.

PID Controllers for Linear Time-invariant

Continuous-time Systems

Figure 1. Feedback control system.

whereandare the proportional, integral and derivative gains respectively. Plant transfer function

where , are polynomials in the Laplace variable . The closed-loop characteristic polynomial is

Stabilization Problem: Determine the values ofandfor which the closed-loop characteristic polynomial is Hurwitz, that is, has all its roots in the open left half plane.

The standard signum function is defined by

Then and denote the numbers of roots of in the open LHP and RHP respectively

The even-odd decomposition of is defined as

where and are the components of made up of even and odd powers of respectively.

Basic Idea: Can determine the root distribution of from knowledge of the the zeros of the odd part and the signs of the even part evaluated at these zeros.

Using the even-odd decomposition of , define

Let , be the degrees of and respectively. To achieve parameter separation multiply by to obtain

- is Hurwitz if and only if has exactly the same number of closed RHP zeros as

- has all three parameters appearing in both the even and odd parts, but the test polynomial exhibits parameter separation, that is, appears in the odd part only while and appear in the even part only. This facilitates the application of root counting formulas to .

For stability must have exactly the same number of RHP roots as . For this a necessary condition is that

has at least

real, nonnegative, distinct roots of odd multiplicity. The ranges of satisfying this condition are called allowable. Let

denote the real, nonnegative distinct roots of of odd multiplicity, and with write

It can be shown using the root counting results mentioned earlier, that the stability condition reduces to:

and therefore the string of integers will be called admissible if it satisfies the above condition

PID Stabilization Algorithm For LTI Plants:

Step 1: For the given and , compute the corresponding , , , and

Step 2: Determine the allowable ranges of . The resulting ranges of are the only ranges of for which stabilizing values may exist;

Step 3: If there is no satisfying Step 2 then output NO SOLUTION and EXIT;

Step 4: Initialize and ;

Step 5: Pick a range in and initialize ;

Step 6: Pick the number of grid points and set ;

Step 7: Increase as follows: . If then GOTOStep 14;

Step 8: For fixed in Step 7, solve for the real, non-negative, distinct finite zeros of with odd multiplicities and denote them by . Also define ;

Step 9: Construct sequences of numbers as follows:

(i) If for some , then define

With defined in this way, define the set as

Step 10: Determine the admissible strings in from (??). If there is no admissible string then GOTO Step 7;

Step 11: For an admissible string , determine the set of values that simultaneously satisfy the following string of linear inequalities:

Step 12: Repeat Step 11 for all admissible strings to obtain the corresponding admissible sets . The set of all stabilizing values corresponding to the fixed is then given by

Step 13:GOTO Step 7

Step 14: Set and . If GOTO STEP 5; else, terminate the algorithm.

Consider the problem of determining stabilizing PID gains for the plant where

The closed-loop characteristic polynomial is

Thus and . Also

and

Therefore we obtain

so that

where

In Step 2, the range of such that has at least real, non-negative, distinct, finite zeros with odd multiplicities was determined to be which is the allowable range. Now for a fixed , for instance , we have

Then the real, non-negative, distinct finite zeros of with odd multiplicities are

Also define . Since which is even, and and ,

and

it follows from Step 10 that every admissible string

must satisfy

Hence the admissible strings are

From Step 11, for it follows that the stabilizing values corresponding to must satisfy the string of inequalities:

Substituting for , , , , and in the above expressions, we obtain

The set of values of for which above holds can be solved by linear programming and is denoted by . For , we have

The set of values of for which the above holds can also be solved by linear programming and is denoted by .

Then, the stabilizing set of values when is given by

The set and the corresponding and are shown in Fig. 2.

Figure 2. The stabilizing set of values when .

By sweeping over different values within the interval and repeating the above procedure at each stage, we can generate the set of stabilizing values. This set is shown in Fig. 3.

Figure 3. The stabilizing set of values.

PID Controllers for Discrete-time Systems

Plant

where and are polynomials in the forward shift operator .

The discrete-time PID controller is given by:

-domain PID controller

-domain closed loop characteristic polynomial:

and Hurwitz stability of this polynomial is equivalent to stability of the original discrete time system. It is clear that we can now proceed as in the previous section

Using the bilinear transformation, we obtain the -domain plant

where

Fig. 4 shows the stabilizing regions in the space of ( , , ) determined using the procedure outlined above.

Figure 4. The stabilizing region in the space of ( , , ).

PID Controllers for Continuous-time First Order Systems with Time Delay

We consider the feedback system of Figure 1 where the plant is described by

represents the steady-state gain of the plant, the time delay, and the time constant of the plant. The controller is of the PID type,

The objective is to determine the set of controller parameters

( , , ). for which the closed-loop system is stable. A complete solution to this problem has been obtained last year. We provide a brief summary of these results.

In this case . Furthermore, we make the standing assumption that and .

Theorem

The range of values for which a given open-loop stable plant, with transfer function as in (??), continues to have closed loop stability with a PID controller in the loop is given by

where is the solution of the equation

in the interval . For values outside this range, there are no stabilizing PID controllers. The complete stabilizing region is given by: (see Fig. 5)

- For each the cross-section of the stabilizing region in the space is the trapezoid T.

2. For , the cross-section of the stabilizing region in the space is the triangle .

- For each , the cross-
- section of the stabilizing region in the space is the quadrilateral Q.

The parameters necessary for determining the boundaries of T, and Q can be determined using

where are the real, positive solutions of

arranged in ascending order of magnitude.

Figure 5. The stabilizing region of for:

[B] Open-Loop Unstable Plant In this case Theorem

A necessary and sufficient condition for the existence of a stabilizing PID controller for the open-loop unstable plant (??) is . If this condition is satisfied, then the range of values for which a given open-loop unstable plant, with transfer function can be stabilized using a PID controller is given by

where is the solution of the equation

in the interval . In the special case of , we have

For values outside this range, there are no stabilizing PID controllers. Moreover, the complete stabilizing region is characterized by: (see Fig. 6)

For each , the cross-section of the stabilizing region in the space is the quadrilateral Q.

The parameters and , necessary for determining the boundary of Q are as defined in the statement of previous Theorem

Figure 6. The stabilizing region of ( , ) for .

PID Stabilization Algorithm for Time-Delay Plants:

- Initialize and , where is the desired number of grid points.

2) Increase as follows: .

3) If then go to Step 4. Else, terminate the algorithm.

4) Find the roots and

5) Compute the parameters and , associated with the previously found by using (1) and (1).

6) Determine the stabilizing region in the - space using Fig. 5.

7) Go to Step 2.

Consider the PID stabilization problem for a plant described by the differential equation

·This process can also be described by the transfer function with the following parameters: , sec, and sec. We use the Theorem above to find the range of values for which a solution to the PID stabilization problem exists.

·We first compute the parameter satisfying the following equation

·Solving this equation we obtain . Thus, from (??) the range of values is given by

·We now sweep over the above range of values and determine the stabilizing set of values at each stage using the previous algorithm. These regions are sketched in Fig. 7.

· Any PID gains selected from these regions will result in closed-loop stability and any gains outside will result in instability.

Now, consider the following performance specifications:

1.Settling time secs;

Figure 7. The stabilizing region of ( , , ) values for the PID controller in Example ??.

2. Overshoot .

We can obtain the transient responses of the closed-loop system for the

( , , ) values inside the regions depicted in the Fig. In general we also need some tolerance around the controller parameters, that is we want the controller to be controller-robust or non-fragile. Thus we only consider PID gains lying inside the following box defined in the parameter space:

By searching over this box, several ( , , ) values are found to meet the desired performance specifications. We arbitrarily set the controller parameters to: ,

, . Fig. below shows the stepresponse of the resulting closed-loop

system. It is clear from the figure that the closed-loop system is stable, the output tracks the step input signal and the performance specifications are met.

Figure 8. Time response of the closed-loop system for Example

The figure also shows the responses of the closed-loop systems for the case of a PID controller designed using the Cohen-Coon method and the Ziegler-Nichols method Notice that in these cases also the system is stable and achieves setpoint following. However, the responses are much more oscillatory.

PID Controller Design for Performance Specs

·In many situations control system performance can be specified by a frequency domain inequality or equivalently an norm constraint on a closed loop transfer function :

·It has been shown by us that the above condition is equivalent to Hurwitz stability of the complex polynomial family:

·In our PID design problem the polynomials will have the PID gains embedded in them and the set of parameters achieving specifications is given by those achieving simultaneously the stabilization of the complex polynomial family as well as the real closed loop characteristic polynomial. It turns out that the set of PID gains achieving stabilization of a complex polynomial family and therefore attaining the specifications can be found by an extension of the algorithm given for the real case.

·

where and are given complex polynomials. The results on PID stabilization presented earlier have been extended to this complex stabilization problem (details omitted)

Consider the following closed-loop transfer functions: considered:

· The sensitivity function:

· The complementary sensitivity function:

· The input sensitivity function:

· Various performance and robustness specifications can be captured by using the norm of weighted versions of the transfer functions. When is a PID controller, the transfer functions (?)-(?) can all be represented in the following general form:

For the transfer function and a given number , the standard performance specification usually takes the form:

where is a stable frequency-dependent weighting function that is selected to capture the desired design objectives at hand.

Define the polynomials and as follows:

and

The performance problem reduces to the simultaneous satisfaction of the following conditions:

1. is Hurwitz;

2. is Hurwitz for all in ;

3.

The above equivalence can be used to determine stabilizing values such that the -norm of a certain closed-loop transfer function is less than a prescribed level. This is illustrated using the following example.

Consider the plant where

We consider the problem of determining all stabilizing PID gain values for which

where is the complementary sensitivity function:

and

1. is Hurwitz;

2. is Hurwitz for all in ;

3.

These sets are shown in the following:

Figure 9: The set

Figure 11: The set of stabilizing values for which

Consider the plant where

Then the sensitivity function and complementary sensitivity function are:

The weighting functions are chosen as: and . We know that stabilizing values meeting the robust performance specification exist if and only if the following conditions hold:

Example: Robust Performance – Cont.

1. is Hurwitz;

2.

is Hurwitz for all and for all ;

3.

The procedure for determining the set of values satisfying conditions (1), (2) and (3) is similar to that presented in the previous example.

Figure 12: The set of values for which

PID Controller Design with Guaranteed Gain and Phase Margins

We consider the problem of designing PID controllers that achieve pre-specified gain and phase margins for a given plant. Let and denote the desired upper gain and phase margins respectively. From the definitions of the upper gain and phase margins, it follows that the PID gain values achieving gain margin and phase margin must satisfy the following conditions:

1. is Hurwitz for all and

2. is Hurwitz for all

Thus the problem to be solved is reduced to the problem of simultaneous stabilization of two families of polynomials.

Consider the plant where

In this example, we consider the problem of determining all gain values that provide a gain margin and a phase margin . A given set of values will meet these specifications if and only if the following conditions hold:

1. is Hurwitz for all ;

2. is Hurwitz for all .

Again, the procedure for determining the set of values is similar to that presented before. The resulting set is sketched in Fig. 13.

Figure 13: The set of values for which the resulting closed loop system achieves a gain margin and a phase margin .

· Similar results have been obtained for first order (lead/lag) controllers

· Extensions to arbitrary fixed order controllers under study

· Software development planned

· Time domain specs. to be incorporated

· Extension to MIMO systems

· Applications

- · Astrom K. J. Åström and T. Hägglund, PID Controllers: Theory, Design, and Tuning, Instrument Society of America, North Carolina, 1995.
- · A. Datta, M. T. Ho and S. P. Bhattacharyya, Structure and Synthesis of PID Controllers, Springer-Verlag, 2000.
- · H. Xu, A. Datta and S. P. Bhattacharyya, “Computation of All Stabilizing PID Gains for Digital Control Systems,” IEEE Transactions on Automatic Control, Vol. AC-46, No. 4, 647-652, April 2001.
- · G. J. Silva, A. Datta and S. P. Bhattacharyya, “New Results on the Synthesis of PID Controllers,” IEEE Transactions on Automatic Control, Vol. 47, No. 2, 241-252, February 2002.
- L.H.Keel, J.I.Rego and S.P.Bhattacharyya, “A New Approach to Digital PID Controller Design" IEEE Trans. Aut. Contr. Vol. AC-48(4), pp.687-692, April 2003.

Motivation

- Design of low order controllers
- Whether there exists a stabilizing controller of a given order
- If so, what is the complete set of stabilizing controllers of a given order
- In terms of a given performance metric, is there a stabilizing controller that achieves a desired performance?
- If there are several performance metrics, what is the set of stabilizing controllers that meet performance objectives in terms of these metrics.

Motivation

- Design of decentralized controllers for a collection of vehicles with a given information structure:
- What are the set of stabilizing controllers that achieve a certain performance (e.g. spacing error attenuation)?
- Such requirements are important for ``scalability of stability’’ for arbitrarily large collections

Problem Statement

Given Data (polynomials):

Questions:

- Do there exist controller gains

such that the following polynomial is Hurwitz?

- If so, what is the set of all such controller gains that make the above polynomial Hurwitz?

Construction of the set of stabilizing controllers

Hermite-Biehler (HB) Theorem: Let

P(s,K) is Hurwitz iff

- The constant coefficients of even and odd polynomials are of the same sign
- Roots of the even and odd polynomial are real and interlace

Construction of stabilizing sets

There exists a stabilizing K (meaning P(s,K) is Hurwitz) iff

- All coefficients of P(s,K) are of the same sign
- there exist (n-1) frequencies

such that

How many LPs for a set of frequencies?

Given a set of (n-1) frequencies

Only the following two LPs must be checked:

What do the feasible sets of LPs mean?

Given a set of (n-1) frequencies,

The two LPs associated with these frequencies indicate all controllers, K, which stabilize the plant and

- which place roots of and alternatively in the disjoint intervals
- Which ensure that all coefficients of P(s,K) are of the same sign

Construction of the set of all stabilizing controllers

One can identify, in a unique way, any feasible LP with

- A set of (n-1) increasing frequencies
- a binary number which indicates the sign of the coefficients
- Problem of determining the set of controllers can therefore be reduced to the search for all these n real numbers
- Store all such sets of n numbers for purposes of design
- Compactify (0, ) (0, 1) using

Example

Even and odd polynomials:

Example

There must exist two frequencies: so that

Example

There should exist such that

Example - Results

Partition (0,1) and search over the partition

Example - Results

Partition (0,1) and search over the partition

Example - Results

Partition

Relevant results from recent literature

Hermite-Biehler theorem is used to

- Synthesize PID controllers (by Bhattacharyya, Keel, Datta, Ho etc)
- obtain a parametrization of all Hurwitz polynomials of a given degree by Djaferis et.al (May 2003, IEEE TAC).
- It maps the interior of a non-negative monotone cone of n frequencies into the set of all monic Hurwitz polynomials of degree n in a bijective manner

Purpose of Outer Approximation

Theorem: If the set of strictly proper controllers of order r is bounded (but not empty), then r is the minimal order of stabilization

Boundedness of an outer approximation of the set of stabilizing controllers of a given order ensures that one is working with minimal order of stabilization

Outer Approximation

Descartes’ Rule of signs:

- The number of sign changes in the coefficients of a polynomial are greater than or equal to the number of its real, positive roots and the difference is always even.
- If all roots of a polynomial are real, then the number of sign changes in the coefficients is exactly equal to the number of real, positive roots of the polynomial
- Application of Descartes’ rule to P(s,K) is equivalent to requiring all its coefficients be of the same sign (leads to two LPs)

Outer Approximation

A Generalization of Descartes’ rule (due to Poincare):

(See Polya and Szego) Let R(s) be a polynomial.

- The number of sign changes in the coefficients of

monotonically decrease with k

2. As k approaches infinity, the number of sign changes in the coefficients of is exactly equal to the number of real positive roots of R(s)

Sketch of the methodology

Applying the Generalization of Descartes’ rule (due to Poincare): It is necessary for P(s,K) to be Hurwitz that

- For every + integer k, the number of sign changes of

is the same as the degree of the even polynomial

- Similar statement for the odd polynomial

For every k, these two conditions generate a number of LPs; the union of the feasible sets of LPs contains the set of stabilizing controllers (outer approximation)

Idea of the proof of Poincare’s result

For k sufficiently large, the number of sign changes in the coefficients of

is the same as the number of sign changes of the sequence

where

Outer Approximation - Interlacing

For two polynomials to have all real positive roots that interlace, given any real positive number, the difference in the number of roots of the two polynomials to the left of the given number is either 0 or 1 or -1.

Using this fact, one can construct an outer approximation which has the following feature:

If K is destabilizing, there is an iteration (akin to the exponent k in the Poincare’s result) such that the kth and subsequent outer approximants will not include K.

Problems that can be tackled

- Simultaneous stabilization of a discrete number of plants
- Synthesis of controllers with a specified gain margin
- Can be posed as a problem of stabilizing a one-parameter family of plants, where the parameter belongs to a compact set, which can be discretized and converted to the problem above

Relevant results from recent literature

Henrion, Sebek and Kucera: Idea of their scheme:

- They use the result that a polynomial P(s) is Hurwitz iff there exists a Hurwitz polynomial Q(s) of the same degree such that P(s)/Q(s) is SPR.
- Q(s) – central polynomial – guessed.
- Requiring SPR of P(s)/Q(s) is equivalent to requiring another polynomial, with coefficients affine in controller parameters, to be non-negative (Siljak’s result from 1973)
- They use Sum-of-Squares method and solve using LMIs

Relevant results from recent literature

Descartes’ Rule of signs – Underutilized tool

Using the Poincare’s generalization, one can show the following result:

If a plant P(s) does not have real, non-minimum phase zeros, there is a two parameter stabilizing compensator that achieves a non-negative closed loop impulse response (only the order can be quite large)

Conclusions

- Presented a method to construct the set of all stabilizing controllers of fixed order
- Presented a method to get a bound for the set of all stabilizing controllers of fixed order
- Both approximations exploit interlacing property of Hurwitz polynomials as described by the Hermite-Biehler Theorem
- The proposed methodology has limitations –

it does not address the question of existence of stabilizing controllers in a crisp manner

Work In Progress

- Synthesis of the set of controllers that acheive a specified performance
- Synthesis of fixed structure controllers for MIMO systems
- Synthesis of controllers which are based on the empirical response data of the plant
- Experiments

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