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ADVANCE IN CONTROL SYSTEM DESIGN AND IMPLEMENTATION WITH MODERN COMPUTING TECHNOLOGY* Yun Li Centre for Systems and Control, & Department of Electronics and Electrical Engineering, University of Glasgow, Rankine Building, Glasgow G12 8LT, United Kingdom E-Mail:
[email protected]
ABSTRACT The most significant impact of modern computing technology on control engineering has been in (1) computational intelligence applied to the design of control systems and to system identification and (2) parallel processing embedded for high-performance implementations of the designed system. This paper develops a uniform and automated methodology for system identification, linearisation and design of control systems, using the state-of-the-art evolutionary computing techniques based upon genetic algorithms. It also presents a systolic/wavefront array based heterogeneous parallel architecture implemented by INMOS transputers and related parallel digital signal processors for adaptive control. Keywords: Control system design, System identification, Genetic algorithms, Parallel processing, Evolutionary computing
I.
INTRODUCTION
With rapid development in modern control theory, control engineers have faced an increasing challenge in selecting for their application an appropriate control law from so many that are available. Then they have to meet the challenge facing the design and optimisation of the parameters of the controller for a linearised model of the plant to be controlled. Both the modelling and design tasks can be regarded as multi-dimensional optimisation tasks in a noisy multi-modal space. These tasks are not easily or accurately carried out by traditional analytical or numerical means, since such methods require a “well-behaved” objective function that is usually unavailable in practice. Thus, in order to avoid these difficulties, we shall transform such a problem into an analysis problem initially, as analysis methods have been well established. Modern computer-aided design (CAD) tools have also reached the stage capable of simulating and evaluating practical engineering systems. These have made it possible to use a genetic algorithm (GA) to “intelligently” interface with such CAD packages for highly accurate modelling and automated designs. The algorithm will be illustrated in Section II. Examples illustrating its use for identification, linearisation and design of control systems will be developed in Section III.
*
This work is supported in part by The Nuffield Foundation, London, under grant SCI/180/91/381/G and by the Engineering and Physical Sciences Research Council of the United Kingdom under grant GR/K24987. The paper is based on an earlier paper (Ref. 10) presented at The Second Asia-Pacific Conference on Control and Measurement, ChongQing, China, June 1995, and another (Ref. 4) presented at The First IFAC Youth Automation Conference, Beijing, China, August, 1995.
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Another challenge the control engineers have encountered is to achieve a high-performance implementation following their successful design. The implementation issues concerning modern computing technology are discussed in Section IV, where the use of systolic/wavefront array processors and transputer supervised parallel digital signal processor (DSP) based heterogeneous architectures will be illustrated. Finally, discussions, conclusions and future work are highlighted in Section V.
II.
THE DESIGN METHODOLOGY
2.1 The problem of design and identification Without loss of generality in the illustrations, consider a second-order linear time-invariant (LTI) system approximated by a transfer function model given by:
G( s) =
a2 s 2 + a1s + a0 b2 s 2 + b1s + b0
(1)
Then the model approximation and identification task is to estimate the six parameters, i.e., ai and bi , ∀ i∈{0, 1, 2}. Here it is assumed that ai ≠0 and bi ≠0 in order to include all the possibilities that may arise from the identification. In linear control system design, this structure can also be used to represent a controller. The design task is thus to optimise the parameters of (1) such that they best meet the design objectives. DEFINITION 1 In the context of design (or modelling), a candidate system can be represented by a uniform parametric vector given by:
{
}
Pi = p1 , ..., p n ∈ R n
(2)
where i stands for the ith possible design (or model) candidate, n the number of parameters required by the control law (or by the model), pj ∈ R the jth parameter of the ith design (or model) candidate with j ∈ {1, ..., n}, and Rn the n-dimensional real Euclidean space which can be quantised (to contain finite number of possible models or designs). DEFINITION 2 The solution space of a control system design (or modelling) problem is given by:
S=
{ P , ∀i i
}
p j ∈ R and j ∈{1,..., n} ⊆ R n .
(3)
DEFINITION 3 The performance index of a control system design (or modelling) is represented by a function f(Pi): Rn→R+ with respect to the design (or modelling) requirements or specifications, where R+ is the nonnegative real space. In addition, the performance index of a design needs to reflect the following design criteria in the presence of practical system constraints: (1) (2) (3) (4) (5)
An excellent transient response in terms of rise-time, overshoots and settling-time; An excellent steady-state response in terms of small steady-state errors; Acceptable stability margins; Robustness in terms of disturbance rejection; and Robustness in terms of parameter sensitivity.
DEFINITION 4 A control system design (or modelling) problem is equivalent to the problem of finding a Po such that: Po = Po ∈ S
f ( Po ) = sup f ( Pi ) . ∀ Pi ∈S
(4)
Traditionally, an optimised parameter identification or control system design task is carried out by conventional calculus based methods. The condition of using these methods is that there exist vector derivatives of the objective index in the multi-dimensional space. This condition is almost impossible to meet in practice1~3. In
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such context, it can be concluded that many optimal control system design and modelling problems are unsolvable problems. On contrast, their related analysis problems are solvable in the numerical domain, once the parameters are given. The following question is, however, left unanswered: Are practical control system design (or modelling) problems solvable in the numerical domain? 2.2 Exhaustive search technique Since conventional numerical optimisation techniques, such as the least mean-squares or maximum likelihood techniques, are based on the Newton or Fibonacci type of “gradient guidance” methods1~3, they are not capable of carrying out practical design and modelling tasks with respect to optimised solutions. This is mainly due to the following problems facing the conventional techniques: (1) Multi-Objective Problem: Conventional optimisation techniques can usually deal with one objective at a time. In engineering practice, there are usually multiple design objectives that may not best be weighted to form a single composite objective; (2) Existence Problem: Gradient guidance can adjust Pi only when ∇f(Pi) exist or the objective functions have well-defined smooth slopes1; (3) Practical Problem: Conventional techniques are almost impossible to work with hard constraint conditions found in practical applications. These constraints include direct domain constraints (such as parameter range requirements and fixed relationships) and indirect inequalities (such as voltage or current limits and other hard nonlinearities). Nor do they work properly in a usually noisy search space1 in practical applications; (4) Multi-Modal Problem: Sequential guiding usually leads to a local optimum1, although parallelism may overcome this to a certain extent. Conventional parallelism, however, includes no effective mechanism to exchange information among parallel search points; (5) A-Priori Problem: It is difficult to incorporate knowledge and experience that a designer may have on the design. In addition, a modern paradigm for control system CAD should also meet the following challenges: (1) (2) (3) (4) (5)
Complexity of practical systems; Required high quality and accuracy of design and modelling; Speed of design (or modelling); Competition with available tools (in terms of ease of use and manual cost, for example); and Robustness, reliability and safety arising from the design and modelling.
It is found that most existing CAD systems cannot meet these challenges easily, due to the above problems of conventional techniques. Using such a CAD package to design, for instance, a designer needs heuristic simulations. He/She has first to input certain a-priori controller parameters, such as those obtained from some preliminary analysis, and should then undertake simulations and manual evaluations using the package. If the simulated performance of the “designed” control system does not meet the specifications as those outlined in Section 2.1, the designer would modify the values of the parameters randomly or guided by his/her real-time gained experience. The engineer would then run the simulations repeatedly until a “satisfactory” design emerges. Clearly, such a design technique suffers from: (1) That the design process is not automated; (2) That the design cannot be carried out easily, since mutual interactions among multiple parameters are hard to predict (i.e., multi-dimensional problem); and (3) That the resulting “satisfactory” design does not necessarily offer the best or near-best performance (multi-modal problem). However, when parameters of a potential system are given, its analysis problem is solvable and encounters no difficulties such as those facing conventional optimisation. One approach to achieving a solvable and possibly automated design could thus be to exhaustively evaluate in S all the possible design choices. Every parameter in (2) may, for example, be quantised into 128 candidate values within its possible range and be encoded by a 7-bit binary string. There are four parameters in (1) that are to be identified, if the system is known to be a causal second-order, and only second-order, system, since in this case b2 can be normalised.
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Exhaustive search by varying the value of every bit of the string will then allow evaluations and comparisons to be carried out on every candidate parameter set. At the end of the enumeration, this would automatically reveal the best (optimised) values of all controller coefficients (or model parameters) without using any optimisation tools. Suppose, however, that each evaluation on one parameter set using a very fast Intel Pentium processor (a main processor paralleled with an on-chip numeric co-processor) takes a typical period of 0.1 second. This task would then take 0.1x1284 sec = 0.1x228 sec = 10 months to complete, which is not practically acceptable. Clearly, the major drawback of such exponential method is that it utilises no search information gained at interim stages. It is reported that the majority of science and engineering problems are potentially NP-complete problems in computing science4. These are the problems that cannot be solved by any deterministic algorithms in polynomial time, but can be solved by a Nondeterministic algorithm in Polynomial time. With “intelligent” information exchange mechanism incorporated in the search process, a genetic algorithm is just such an algorithm that can transform an exponential problem to an NP-complete problem. 2.3 Genetic algorithm based evolutionary computing technique Emulating Darwin’s evolutionary principle of “survival-of-the-fittest” in natural selection and genetics, the genetic algorithm1,2 has been found very effective and powerful in searching poorly understood, irregular and complex spaces for optimisation and machine learning. This algorithm and its associated genetic programming3, evolutionary programming and evolution strategy techniques are referred to as “evolutionary computing” techniques. Supported by the Schema Theory1-3, it has been shown that a genetic algorithm offers an exponentially reduced search time. These techniques have received a rapidly growing international attention and have been successfully applied to systems and control engineering problems4, including design of simple PID5,6, linear7~10, optimal5,11~13, adaptive14, robust15, sliding mode16,17, fuzzy logic18~20, and neural network21~23 control systems, parameter estimation and system identification24~26, linearisation25 and controller order reduction 27. A typical evolution process of the GA is shown in Fig. 1. Consider again binary strings representing a design or modelling. Such a string is termed a “chromosome” in the GA context and a bit of the string is termed a “gene”. A GA uses three basic operators termed selection, crossover and mutation. Selection is used once the initial population involving, usually, a fixed number of chromosomes representing the parameter candidates is formed. As a result of selection, a new generation of population is evolved based on their fitness. Here the fitness is a measure of relatively how well a candidate parameter set meets the accuracy specifications, as given by Definition 3, and dictates the probability of that set to reproduce and survive in evolution. The task of evaluating fitness can be carried out by a conventional CAD simulation package, which is interfaced with the GA. The crossover operator is used to produce off-spring that inherit a portion of the search information of their parents. In the mean time, mutation plays a secondary role in the GA to alter the value of a gene at a random position in the chromosome string, discovering new or restoring lost genetic material. This serves to keep the diversity in the population and searches a neighbouring point. By this stage, a new generation is formed and then the process repeats itself until the fittest design emerges, as schematically depicted by Fig. 1.
III.
SYSTEM IDENTIFICATION AND DESIGN EXAMPLES
3.1 System identification and linearisation This sub-section shows how the genetic algorithm is applied to the identification of an LTI DC servo-system. The typical differential equation defining the open-loop servomotor system with field control is given by:
d 2ω JR + LB dω RB K + + ω = T vin LJ dt LJ dt LJ
(5)
where vin∈[-5V, 5V] is the input control voltage as an indirect constraint, KT in Nm/A the torque constant, R in Ω the resistance of the motor winding, L in H the inductance, B in Nms the friction coefficient of the shaft, and J in Kgm2 the moment of inertia of the load and the machine. The transfer function is in the form of (1) with a2=0 and b2=1.
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All parameters in one candidate set can each be coded by a 7-bit string for a resolution of 128 within the specified range of their possible values. Note that, however, other coding methods and bases exist8,16,20 and that logic terms can also be encoded by a numeric string22. Here the population size is fixed at sp=50, in view of the complexity of the problem. The initial population can be generated randomly, if no a-priori knowledge is available, and should include known estimations for a faster convergence. Final optimised designs
Initial/random designs coded
Selection
f(P1: 1 1 0 1 0 1 1 0)=5% f(P2: 1 0 0 1 0 1 1 1)=60% f(P3: 0 1 0 0 1 0 0 1)=35%
f(P3) f(P1) f(P2)
New generation
Decoding, simulation, evaluation
Mutation P2: 1 0 0 1 0 1 1 1 P2’: 1 0 1 1 0 0 0 1 P3’: 0 1 0 0 1 1 1 1
Crossover P2: 1 0 0 1 0 1 1 1 P2: 1 0 0 1 0 1 1 1 P3: 0 1 0 0 1 0 0 1
Conventional CAD
Figure 1. Evolution of coded designs by fitness evaluation based genetic operations. In GA based identification, the excitation input to the real system and the candidate model can be a simple unit step. The accuracy of the model being identified is then indexed by, for example, the sum of absolute errors (L1 norm) between the actual output of the system and the simulated output of the model, as given by: N
esys ( Pi ) = ∑ ω j − ω$ j
(6)
j =1
where N=1000 is the total number of data. Clearly, the performance index can also be measured by an L2 or L∞ norm, as they and L2 are finitely bounded with one another (i.e., metric equivalence)8. Note that non-numerical functions, such as rules or fuzzy logic terms, can also be included in the index. It can also contain terms that better reflect how engineers interpret certain specifications in practice 9. To fill the slices of the roulette wheel shown in Fig. 1 for GA selections, normalised inverse values given by (6) for all candidate models in each generation may be used as fitness. However, such fitness varies with generations and nonlinear scaling4 may have to be used to avoid polarisation and to help with a healthy convergence. Thus the “ranked roulette wheel”8 is developed here with fitness pre-assigned values determined by an arithmetic series, as given by:
f (k) =
2k − 1
∑ k =1 (2k − 1) sp
=
2k − 1 250
(7)
where k∈{1, 2, ..., 50} represents the “rank” of esys of an individual chromosome in the generation. Here a higher rank is assigned to an individual with a lower error. Then the value given by (7) is used as the probability of the chromosome in question to reproduce itself for off-spring. Clearly, other arithmetic series such as k may be used for a larger population size and a geometric series such as 2(2k-1) may be used for a smaller population. Note that there are also other selection mechanisms, such as proportionate selection, tournament selection, elitist strategies, and steady-state selection schemes1~3,17. The crossover is typically applied at a rate of, for example, about 60% of the population and the crossover point for each pair is randomly selected. In addition, this operation can also be applied to multiple random points. On contrast, mutation is applied at a rather low probability of, typically, 0.1%~10% of the entire population. Adaptive mutation schemes can also be introduced to prevent similarities of the parental pair 16. -5-
This procedure for identification of (5) typically runs for about 2 hours on a 50 MHz Intel 80486DX2 machine, when the fitness has approached a constant “mean-error” of 7%. This error is viewed as mainly contributed by the structural uncertainty of the model governed by (5), which ignored the nonlinearity and transport delay of the real physical system. The identified values of parameters in (5) are given by:
JR + LB RB KT = 39.142, = 86.186, = 0.054 LJ LJ LJ
(8)
Note that, in order to reduce the error further, this GA based identification methodology can easily be extended to a more accurate model that incorporates a transport delay term, e-sT, and include T in the parameter set. Similarly, this method can also be applied to approximating a “completely” nonlinear system by an LTI model25. Here a high order transfer function with the structure of (1) can be used to approach the nonlinear system around the equilibrium point or with input excitations restricted within the operating ranges. Direct nonlinear system identification can also be carried out25. 3.2 Automated design of a uniform linear control system Based on the identified model, this sub-section demonstrates how a GA can be used to automate the design of a performance based, as opposed to control scheme based, linear controller. When the model is known, a control engineer would usually select a control scheme (such as PID, phase lead-lag, pole-placement, optimal LQG, H∞ or µ-synthesis) for design, trying to make the choice as “wise” as possible. Such pre-selected control scheme may not, however, best suit the application at hand. It can be seen that, in implementations, all of the control schemes mentioned above are of the form of the LTI transfer function described by (4). It is thus unnecessary to have to select a specific control scheme before the design process takes place. This view forms the underlying theme of the performance-based GA automated design methodology developed here. Similar to the identification process, it is necessary to establish for all possible candidate coefficient sets a global measure of fitness as given by (7). A simple design error function that reflects steady-state and transient responses is given by8,16: N
{
edesign = ∑ e j + ∆e j j =1
}j
(9)
Here ej is the closed-loop error at simulation step j and ∆ej the change of error from the previous step. The simple linearly-forgetting weighted cost will better reflect the rise time, amount of overshoot, oscillations and stability. This form of performance-based design also avoids the need of studies on asymptotic stability, existence or convergence. Note that, however, it is always application-specific in defining error cost. For example, |ei | and |∆ei | in (9) could be weighted distinctively in a similar way to H∞ and µ-synthesis based control. Response shape based specification can also be used with penalties9 and multi-objectives may be used separately12. However, the task of intelligent selection of fitness functions that best reflect the designer’s needs may be performed by the GA as well, the topic of which is also studied at Glasgow. The GA is applied to the design of a second-order uniform linear controller for the DC servo-mechanism identified by (5) and (8). It took about 20 minutes in average to converge. The transfer function of the controller is given by:
H (s ) =
19.28s 2 + 121.66s + 108.84 s 2 + 72.77s + 0.14
(10)
Its response to a step input of 60 rpm (after a 9:1 step-down gear-box) is shown in Fig. 2. It tends to offer a rapid response with small errors. Note from (10) that the GA attempted to incorporate an integrator to convert the Type 0 plant given by (5) to a Type 1 system for small steady-state errors. Clearly, this design example is a simple one. It does, however, demonstrate the capability of the GA. The method can be used for designing any LTI controllers that are of a higher order7. Further, the GA assisted performance based uniform design methodology eliminates the artificial division between “classical control” and “modern control”. -6-
Figure 2. Step response of (11) on the plant model given by (5) and (8)
IV.
PARALLEL IMPLEMENTATIONS
In addition to the computational intelligence features of modern computing technology, parallel processing, in various architectures and networks, has been the other major advancement in the past few decades. It offers a genuine hope for high-performance and high-accuracy real-time implementations of advanced control systems28~30, such as adaptive control systems. Since the implementation of modern real-time control systems involves heavy computations that must be completed within one sampling period (or one control cycle), prolonged processing time sacrifices the system ability in dealing with high dynamics and forces the control algorithms to be simplified, with consequent loss of performance. Further, the implementation requirements imposed on parallel processing for feedback control are more strict than those on high-performance signal processing applications where only the throughput rate is the primary concern. In control systems, however, both the throughput and processing delay (measured by control cycles) are important issues, due to the critical involvement of feedback. Prolonged latency reduces the stability phase margin of the overall system in the same way as pure transport delay in the plant. The VLSI based systolic/wavefront array architectures that meet the throughput and latency requirement for embedded control systems have been developed28~30. Fig. 3 shows one of the simplest architectures, a semisystolic array28~30, implemented on A100 parallel DSPs with a heterogeneous architecture supervised by a transputer. Here, the designed controller coefficients, as well as the calculated error signal from the sampled output of the plant, are fed by the T212 transputer to the A100 arrays.
Figure 3. Transputer/A100 based heterogeneous implementation of GA based adaptive control system A step towards autonomous control systems is, however, adaptive control, where the design is amalgamated with the implementation. This is in conjunction with the uniform design methodology developed. Here, in addition to the A100 device used for the implementation, INMOS transputers are used to perform the combined identification and design tasks running the GA. This yields the overall adaptive architecture schematically depicted by Fig. 4, which belongs to a multiple-instruction multiple-data (MIMD) architecture. Note that the design and coefficient updating phase is run at a much slower rate than the control signal provision30, since the -7-
plant parameters vary much slowly compared with the plant dynamics. This heterogeneous system is capable of delivering control signals at a rate of 47 KHz30, which is regarded as high enough for many engineering applications. Coefficients updates in the cells
GA Command
Output
Plant
+ -
IIR cell pipeline
Figure 4. Heterogeneous architecture for GA based adaptive control
V.
CONCLUSIONS AND FURTHER WORK
This paper has developed a uniform methodology for system identification and controller design, using the GA based evolutionary computing technique. Enhancement of this method and design of autonomous system out of plain plant I/O data are currently studied at Glasgow, together with related projects in GA automated µsynthesis, multivariable control, and modern electric machine and drive design. The developed methodology not only avoids the tedious manual trial-and-error process arising from the lack of tractable analytical and numerical design approaches, but also yields control systems that give a better performance than manual designs in terms of transient and steady-state performance. The ease of design is not, however, achieved at no cost. One trade-off of accuracy is the quantisation error in coding, which however may be ignored. The other arises from the improbability of finding the “exact” global optimum due to limitations of the nondeterministic GA search. However, the genetic algorithm based approach allows multiple and complicated objectives and practical limitations to be imposed, in addition to offering an exponentially increased speed that makes design automation possible. The heterogeneous architecture presented in this paper provides a high throughput rate and a short processing delay for advanced control systems. Since a genetic algorithm works on the entire population at multiple searching points simultaneously, as opposed to one individual point at a time, it is thus best suited for parallelisation. The computational intelligence and parallel processing features in the system offer a genuine hope for the next generation of integrated control systems - the completely autonomous computer-control system directly incorporated in system manufacturing.
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