Multiobjective Optimization of an Operational Amplifier by the Ant Colony Optimisation Algorithm

The Ant Colony Optimization (ACO) algorithm is used as a multi-objective optimization technique to size a most popular analog circuit, the CMOS operational amplifier (Op-Amp). The work consist of finding the more convenient transistors sizes, including the channel widths and lengths, in order to meet or reach the specified requirements such as the voltage gain Av, the Common Mode Rejection Ratio CMRR, the die area A, the power consumption P and the Slew Rate SR. SPICE simulations are used to strengthen and to validate the obtained sizing/performances.


Introduction
Over the past decade, significant progress has been realized with the appearance of a new generation of powerful and approximate optimization methods, known as metaheuristics [1]. Such methods are used to solve real-world problems within a reasonable amount of time. They always offer 'good' approximat ion of the 'best' solutions for optimization problems [2]. One of the incoming problems to be resolved in the nearer future is the sizing of electronic circu its given the continuous increase of the integration densities. So, the designers and electronic engineers had a good and exciting challenge to reach, that is to find a technique, wh ich can easily determine the components sizing, taken into account a well determined specifications. So me (meta-) heuristics were proposed in the literature and were used by some designers to optimize the sizing of the analog co mponents automatically, such as Tabu Search [3,4], Genetic A lgorith ms (GA) [5], local search (LS) [6], Wasp Nets (WN) [7], Bacterial Foraging Optimization (BFO) [8], Particle Swarm Optimization (PSO) [9] and recently Ant Co lony Optimizat ion (ACO) [10,11].
These algorithms have often dealt with single-object ive optimizations; however, the optimizat ion of analog circu its is generally a mu lti-objective problem. They are always formed by at least two conflicting performance functions.
That means th at imp rov ing on e perfo rmance resu lts automatically into the degradation of another one. In this way a set of several meta-heuristics algorith ms have been developed, such as Mult i-ob jective Optimizat ion Genet ic Algorith m (MOGA) [12], Mu lti-object ive Optimization Particle Swarm Optimization (M OPSO) [13].
In the domain of meta-heuristic methods, an important interest has been paid to the Ant Colony Optimization algorith m. The basic idea is to imitate the cooperative behaviour of ant colonies in order to solve combinatorial optimization problems within a reasonable amount of time, The A CO is actually recognized as one of the most successful strands of swarm intelligence [14]. So me ACO-based algorithms have been proposed such as the multi-object ive optimization problems MOA CO [15], the Mult iple Objective Ant-Q Algorithm (MOA Q) [16], the Pareto-Ant Colony Optimization (P-A CO) [17] and the Ant Algorithm for Bi-criterion Optimization Problems [18].
In a previous work, we have adapted and used the MOACO algorith m for a two objectives electronic circu it optimization namely the Second Generat ion Current Conveyors [19]. At the present we propose to use the same algorith m in order to optimize the sizing of the CM OS Op-A mp, which is a more popular analog circuit requiring many h ighly interdependent performances.
Thus the subsequent section will present an overview of the ACO technique fo llo wed by with a presentation of the proposed adaptation of the ACO technique for analog circuits optimization and introduces the MOACO. The third deals with the optimal sizing of the CMOS Operat ional Amplifier (Op -A mp) and presents the main results and simu lations. Finally, the paper will be concluded in the final section.

Ant Col ony Optimization
The ACO technique is inspired by the collective behavior of deposition and monitoring of some traces as it is observed in insect colonies [20,21], such as ants. It is for examp le well known that ants deposit pheromone on the ground in order to mark some favourable paths that should be follo wed by other members of the colony. Fig. 1 shows an illustration of the ability of ants to find the shortest path between food and their nest. Ants communicate indirectly through dynamic changes in their environ ment (pheromone trails). The ACO was initially used to solve graph related problems, such as the travelling sales man problem [22], vehicle routing problem [23], Optical networks routing [24], and bioinformat ics problems [25]. A graph is composed of vertices and edges. Each ant constructs its own path fro m the starting to the final vertex by ''walking'' along edges connecting the vertices by deposing a certain amount of pheromones (a chemical substance) that evaporates during the time, unless it is reinforced by another ant 'walking' along the same edge. Thus, the 'best', i.e. the shortest, path is determined on the base of these pheromones. Besides, movement of the ants is highly conditioned by their visibility regarding the final objective.
For solving such problems, ants randomly select the vertex to be visited. When an ant k is in the vertex i, the probability for going to the vertex j is given by the following expression [26,27]: Where ij d is the distance between vertices i and j.
The pheromone rate values are updated during each iteration by all the m ants that have built a solution in the iteration itself. The phero mone rate ij τ , which is associated with the edge joining vertices i and j, is updated as follo ws: where ρ is the evaporation rate, m is the number o f ants, and is the quantity of phero mone laid 'deposited, or dropped of'on edge (i, j) by ant k: Q is a constant and L k is the length of the tour constructed by the ant k.
The ACO approach attempts to solve an optimizat ion problem by iterat ing the follo wing two steps: • The Candidate solutions are constructed using a pheromone model, that is, a parameterized probability d istribution over the solution space; • The candidate solutions are used to mod ify the pheromone values in a way that is deemed to bias future sampling toward high quality solutions.

Adaptation
The main idea consists of constructing a graph that imitates the movement of the ants [28,29]. Then, we construct a graph composed of the discretized variab le vectors, corresponding to the graph vertices. Thus, each ant will construct its path by a random displacement fro m a variable value to another, as it is shown in Fig. 2; V1, V2, V3…VN constitute the discrete variable vectors. In short, each ant k will randomly chose a path (values of the Ant Colony Optimisation Algorithm V1, V2 …), accord ing to the probability given by expression (1), and form a non-connected directed graph while randomly generating a rate of phero mone at the formed graph edges. At each iterat ion, the path giving the min imu m value of the objective function (OF) sees its pheromone rate increasing, in contrast with the other paths, for which the pheromone rates start to evaporate with respect to expression (3).
In the mu lti-ob jective problem, we seek to optimize several functions that are usually interdependent. So, the concept of Pareto optimality is used [30,31]. This approach consists, of the following, with n parameters (decision variables) and K objectives: if, and only if:  In the mult i-object ive optimizat ion problem, a set of non-dominated solutions form the Pareto frontier. An examp le is shown in Figure 3, where the solid (filled) circles represent the non-dominated solutions which form the Pareto frontier, wh ile the open circles represent the dominated solutions. This result corresponds to a two-objective optimization problem, where the goal was the minimization of the two objectives, i.e. to search for the non-dominated solutions located along the Pareto frontier.
To resolve the multiobjective problems, we proposed the algorith m shown in Figure 4. In the initialization phase: Ants are generated each starting with a set X, the object ive weights P k is determined randomly for each ant. In the construction phase of the algorithm, each ant tries to construct a feasible set X by using a pseudo-random proportional ru le. After a set has been constructed, its feasibility and efficiency is determined. Phero mone updating is performed by using the best solution X k of the current iteration for each objective k.

Operational Amplifier Optimization
The two-stage CMOS operational amp lifier (Op-A mp) shown in Figure 5, is considered as an examp le for the validation of our proposed algorith m. In fact the design of the Op-Amp continues to pose a challenge as transistor channel lengths scale down with each generation of CMOS technologies [32].
• The Co mmon Mode Rejection Rat io CMRR: • The die Area A : • The Slew Rate SR: Those expressions (5) and (7), were obtained by considering the s mall signal equivalent transistor's models. V dd and V ss are respectively the positive and the negative supply voltages; W 1 -W 8 and L 1 -L 8 are the gates widths and the channels lengths of the transistors M 1 -M 8 respectively. I bias is the bias current, gm refers to the transconductance of the MOS transistor, C ox , λ n , λ p , μ n and μ p are technological parameters. C c is a compensation capacitor and C TL is the total capacitance at the output node which can be exp ressed as: 6 7 TL L C C Cgd Cgd = + + Cgd 6 and Cgd 7 denote to the parasitic grid to drain capacitance for transistor M 6 and M 7 respectively.
Determining the optimal dimensions of the transistors for a specific design involves a tradeoff among all these performance measures. Each transistor must be in saturation. Exp ressions (11)-(14) g ive the corresponding constraints, that have to be satisfied when computing optimal sizes of the transistors M 1 (and M 2 ), M 5 , M 6 and M 7 respectively.
where, V tp and V tn are the PMOS and the NM OS threshold voltages, respectively. This optimizat ion belongs to the family of NP hard problems; in fact, there are 11 design parameters to optimize for the two-stage Op-Amp; the widths and lengths of all tran-sistors, (W 1 -W 8 and L), the bias current I bias and the value of the compensation capacitor C C , in addition to the various constraints of the problem. Note that all the channel length L is considered the same for all the transistors.
The considered optimization problem is a typical mu lti-objective one, consisting of minimizing two objective functions (the die area and the consumed power), and maximizing the other performances.  The MOACO algorith m when applied to our optimizat ion problem, using the parameter values given in Table 1, g ives 43 optimal parameter designs. Table 2 present five of these parameter designs and their corresponding performances. Also, it is to be noted that the computing time equals 184s. SPICE simu lation results performed, using AMS 0.35μm technology, Voltage power supply is V dd /V ss =+2.5V/ -2.5V, are presented in the Tab le 3; they show the good agreement with the expected ones. Figure 6 shows SPICE simu lation results for the five optimal designs, of the gain. The choice between the sets of the determined optimal parameters, given by this MOACO algorith m, will depend on the desiderata of the designer.

Conclusions
The presented work proposes an adaptation of the ant colony optimizat ion technique to the optimal sizing of analog circuits. We show the practical applicab ility of the ACO to optimize performances of electronics integrated circu its and its suitability fo r solving a mu lt iobjective optimization problem. The proposed algorithm is validated by the Operational A mplifier performances optimization. Viability of the technique was proved via SPICE simu lations.