Controller desimn usinm linear programming for svstems with constraints Part 2 Controller design by F. Tadeo and M. J. Grimble

This three-part article reviews the techniques used to design controllers, employing linear programming, that minimise the maximum amplitude of selected signals. These controllers can satisfy design specifications for saturation characteristics on amplitude or rate. It is shown also how given an existing linear controller it is possible to check if, for a given set of typical inputs, the signal would exceed its maximum amplitude or rate. This is very useful from a practical point of view. The first part (December 2001) gave a Tutorial introduction to the topic. This second part reviews controller design.

p to this point, it has been seen that, to test if a control system fulfils given performance specifications, it is only necessary to check a U condition on a norm. The design of optimal

controllers for systems with bounded amplitude signals involves the problem of finding the controller that minimises the peak-to-peak norm of these output signals for every possible input, which is just the controller that minimises the norm of the corresponding transfer- function. More precisely the problem can be stated as:

min IIH(Co)lIi stabilising Co

where Co is any controller that internally stabilises the closed-loop system (that is, that the closed-loop system is stable and there are no polehero cancellations between the controller and the plant). This problem has a solution if certain conditions hold. There follows a brief intro- duction to the calculation of the optimal controller using linear programming.

Solution via linear programming Earlier discussions revealed that many practical

controller design problems can be transformed into a minimisation problem in terms of the I1 norm of certain transfer-functions. This section deals with the trans-

formation of these minimisation problems into linear programming problems, which can be solved using commercially available software. There will be a departure from the usual approach,6 which is based on solving the optimisation problem in the Youla-parameter domain (this approach will be discussed in the third part of this Tutorial, in the section on ‘Further reading’). An alternative formulation will be presented, based on optimising directly the impulse responses taking into account the standard interpolation and feasibility constraints, without using the Youla parameter as an intermediate step. It is believed that this variation of the solution is a more natural way of solving these optimisation problems, for designers not familiar with Youla parameterisations.

In general the method applied to solve a design problem depends on the size of the matrices involved: the problem

is called a one-block problem if H i s square. This problem can be solved more easily than when H is non-square, which is called a multi-block problem. The one-block and multi-block cases will now be discussed.

COMPUTING & CONTROL ENGINEERING JOURNAL FEBRUARY 2001

TROLLER DESIGN

One-block problem The basic idea of the solution is based on optimising

directly the elements of the impulse response of H, with additional constraints to ensure the internal stability (is.) of the feedback system. That is, the controller cannot have any pole that cancels any unstable zero of the plant (or any zero that cancels any unstable pole of the plant). These constraints are called interpolation constraints, which can be easily included as constraints in the accepted impulse responses, and absorbed in the linear programming problem, as is now discussed. From now on, the impulse response of H(z) will be denoted as @, and the elements of the impulse response as {@[k]}T=i. The elements of the (truncated) impulse response are directly calculated by optimisation, as is shown now.

Interpolation constraints Denote by {zh} the plant non-minimum phase zeros

(i.e. zeros outside the unit-circle) and by @ph} the plant unstable poles. For simplicity and without loss of generality, we suppose from now on that they are non- repeated and real. For the feedback system to have internal stability, the controller should not cancel these poles and zeros. This means that Co Wmust take the same value of W at these zeros and poles: Co W must be 0 if it is a zero and w if it is a pole. This condition can easily be transformed into requirements on the characteristic transfer-functions. For example, one of the transfer- functions more frequently used in robust controller design is the so-called sensitivity:

1 S= ~

1 + c o w

which corresponds to the transfer-function between a disturbance on the output in the plant (d) and the plant output (2). Then if H i s the sensitivity, the interpolation constraints are:

Another frequently used transfer-function is the comfile- mentary sensitivity:

T=- cow =cows 1 + c o w

which corresponds to the transfer function from the command input (c) to the plant output (2). The inter- polation constraints for the complementary sensitivity are:

A third transfer-function, frequently used in systems

with constraints in the control signal is the control sensitivity:

which corresponds to the transfer function from the command input (c) to the control signal (u). Following a similar approach, it can be seen that only the unstable poles give interpolation constraints, which are M@h) = 0.

Transformation to a linear programming problem

constraints are equivalent to the requirement: In terms of the impulse response, the interpolation

where @ denotes the impulse response of the transfer- function H to be optimised, @(ah) the evaluation of the transfer function H at z = ah, and hi(ah) is the corres- ponding required value of the transfer-function at this point given by the interpolation constraint (0 , l etc.):

min 11Hll1 His.

0 m&ll@IIi such that @(ah) = H(ah)

Noting 11@1)i =sl@[i]l (from the definition of the h norm) and Zl@[i]lahz = &(ah) (from the properties of the impulse respohse, then the problem is just:

such that

To transform this problem into a standard (infinite dimensional) linear programming problem, it is only necessary to eliminate the absolute value of the cost function. As is common practice in optimisation, change @[i] = @+[i] - @-[i], where @+[i] 2 0 and W [ i ] 2 0. As @+[i] and @[i] appear in the cost-function to be minimised with a positive weight, it is possible to deduce that one of them is zero:

If the optimal @[i] is positive then @-[i] = 0 and

*If the optimal @[i] is negative then @-[i]>O and

If the optimal @[i] is zero then W[i] = 0 and @+[i] = 0.

@+[i] > 0.

@+[i] = 0.

The optimisation problem can then be stated as:

COhPUTING & CONTROL ENGINEERING JOURNAL FEBRUARY 2002

such that

A(@+ - @-) = b (interpolation constraints)

where (@+ - W) is an infinite column vector, A is a n x 00 matrix, n is the number of zeros of U and b is a finite row vector.

This is already a linear programming problem (although with infinite number of variables and con- straints). However, in order to simplify the extension to the multiblock (or multivariable) case, the standard transformation is frequently used

to transform the minimisation problem into the mini- misation of a single variable y, subject to additional constraints, which we will denote as norm constraints:

min y

of inputs and output signals (disturbance, measurement noise on output andor control signals). These trade-offs can be transformed into a parallel optimisation on two characteristic transfer functions, such as the sensitivity, the control sensitivity or the complementary sensitivity. This parallel optimisation problem is usually called a mixed sensitivity problem. The simplest formulation of the mixed sensitivity problem for a SISO system can be given as the following minimisation problem:

The same approach presented to solve the one-block problem can be followed to transform this minimisation problem to a linear programming problem. The only modification is that it is necessary to add additional constraints, called feasibility constraints.

Feasibility constraints Feasibility constraints relate the transfer-functions

in the different blocks of the optimisation (to make sure that the controller optimised is the same in every block). For example, the relation between sensitivity and the complementary sensitivity can be seen to be:

such that: S + T = l

i=l h + [ i ] + $-[i] I y (norm constraints)

A(@+ - @-) = b (interpolation constraints)

This is a semi-infinite linear programming problem (it has infinite variables). Fortunately it can be proved that the optimal solution is finite, of dimension N (an upper bound on its size can be calculated from the dual problem). The original problem can then be truncated to obtain a finite dimensional problem. It is possible to solve these linear programming problems using any of the commonly available tools, such as linear program solvers included in the Matlab Optimization Toolbox, or the EISPACK or NAG libraries. Frequently, the corres- ponding linear programming problem has redundant constraints, which must be eliminated by the linear program solver before solving the linear program. The technique is presented in the third part of this article using an academic example in Matlab.

Multi-block problem It has been shown how to solve the 11 optimisation

problem for one-block problems. However, in most practical problems the transfer-matrix to be optimised is non-square (the number of external input signals is different from the number of external outputs), so it is necessary to extend the method to the so-called multi- block problem. A typical example is the mixed sensitivity problem, where the designer must consider trade-offs in the minimisation of the amplitude of different combinations

The relation between the complementary sensitivity and the control sensitivity can be seen to be:

T = WM

Other relations can be deduced from these. The main difficulty is that these feasibility constraints

lead (theoretically) to an infinite number of constraints in the primal problem, so in practical problems the number of variables (or constraints) is truncated and sub-optimal solutions are obtained (with arbitrary accuracy). In the multi-block case, it is not possible to calculate a general bound of the length of the optimal solution, so it is not possible to know in advance the length of the optimal response.

Transformation to a linear programming problem:

min y, subject to the following constraints: The resulting minimisation problem can be stated as

1 Norm constraints:

i=l h z f [ i ] + z ~ [ i ] I y

2 Interpolation constraints:

COMPUTING & CONTROL ENGINEERING JOURNAL FEBRUARY 2002

Observe that the addition of the feasibility constraints usually makes some of the interpolation constraints redundant. This redundancy means that it is usually worth checking the possible redundancy of the inter- polation constraints, so some of them can be dropped. In the mixed sensitivity problem only one set of constraints is necessary, and the complexity of the problem can be reduced. For example, if the designer is considering a mixed sensitivity problem with the sensitivity and the complementary sensitivity, related by the feasibility constraint S + T = I, it is possible to check that half the interpolation constraints are redundant. If we make the sensitivity equal to zero at the unstable zeros (S(zh) = O), this means that we are implicitly making the complementary sensitivity equal to 1 at the unstable zeros (T(zh) = 1) and vice versa. This means that it is only necessary to consider one of the two sets of interpolation constraints.

An example will be presented in the third part of this article to show how the designer can transform the mixed sensitivity problem in a linear programming problem. Although the length of the optimal impulse response (except in some special cases) is usually unbounded, the designer can calculate a (probably) sub-optimal solution of length m. The resulting minimisation problem is obtained by supposing that @+[i] = W [ i ] = 0 when i 2 m. m can be selected tradingoff optimality of the solution and complexity of the controller (by increasing m a more optimal solution is obtained, but with a longer impulse response).

Controller design So far, we have only discussed the method to calculate

a controller to minimise an 11 norm. The method we propose is based on solving a mixed sensitivity problem that takes into account the different signals in the feedback systems. This mixed sensitivity problem includes weights selected by the designer based on the signal description and other requirements. For instance, in a regulation problem, the input signals involved are (bounded) disturbances; the output signals are the measured output and the control signal. The mixed sensitivity problem to be solved is then:

The designer must select the weights based on the signal description. In this article the proposal is to select the weights in the frequency domain, as upper bounds on the frequency response of the 11 specification, determined from an analysis of the plant model. This technique will be shown in the case study in an article to be submitted to the Journal later.

Reference 6 DAHLEH, M. A,, and DIAZ-BOBILLO, I. J.: 'Control of uncertain systems: a linear programming approach' (Prentice-Hall, Englewood Cliffs, NJ, USA, 1995)

Part three ofthis article (to be published in the next issue) will look at some design examples.

0 IEE 2002 Dr. Fernando Tadeo is with the Dpto. Ingenieria de Sistemas y Automatica, Facultad de Ciencias, 47005 Valladolid, Spain. Tel: +34 983 423566 Fax: +34 983 423161; E-mail: fernando@autom.uva.es. Prof. M. J. Grimble is with Industrial Control Centre, University of Strathclyde, United Kingdom.

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